The Center and Focus Problem: Algebraic Solutions and Hypotheses (Chapman & Hall/CRC Monographs and Research Notes in Mathematics) [1 ed.] 1032017252, 9781032017259

Table of contents :
Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Authors
Introduction
1 Lie Algebra of Operators of Centro-Affine Group Representation in the Coefficient Space of Polynomial Differential Systems
1.1 Two-dimensional Polynomial Differential Systems
1.1.1 Affine System
1.1.2 System with Quadratic Nonlinearities
1.1.3 Quadratic System
1.1.4 System with Cubic Nonlinearities
1.1.5 Cubic System
1.2 One-Parameter Linear Groups of Transformations of the Phase Plane of System (1.1)-(1.2)
1.3 Centro-Affine and Unimodular Transformation Groups of the Phase Plane of System (1.1)-(1.2)
1.4 Lie Operators of One-Parameter Linear Groups and Their Representations in the Coefficient Space of System (1.1)-(1.2)
1.5 Operators of Representation of the Linear Groups (1.12), (1.14), (1.16) and (1.17) in the Space of Variables and Coefficients of System (1.1)-(1.2)
1.6 Lie Algebra of Operators of Centro-Affine Group Represen-tation in the Space of Variables and Coefficients of System (1.1)-(1.2)
1.7 Comments to Chapter One
2 Differential Equations for Centro-Affine Invariants and Comitants of Differential Systems and Their Applications
2.1 Concept of Centro-Affine Comitant and an Invariant of Differential System
2.2 Centro-Affine Transformations of System (1.1) (1.2)
2.3 Differential Equations for Centro-Affine Invariants and Comitants
2.4 Rational Absolute Centro-Affine Invariants and Comitants and Their Applications
2.5 Examples of Algebraic Bases of Centro-Affine Comitants and Invariants for Some Differential Systems
2.6 Comments to Chapter Two
3 Generating Functions and Hilbert Series for Sibirsky Graded Algebras of Comitants and Invariants of Differential Systems
3.1 Formulas for Weights of Centro Affine Comitants and Invariants of Given Type
3.2 Initial form of Generating Function for Centro-Affine Comitants of Differential Systems
3.3 Examples of Reduced Forms of Generating Functions for Centro-Affine Comitants of Differential Systems
3.4 Hilbert Series for Graded Algebras of Unimodular Comitants and Invariants of Differential Systems
3.5 Comments to Chapter Three
4 Hilbert Series for Sibirsky Algebras S[sub(Г)] (SI[sub(Г)] ) and Krull Dimension for Them
4.1 Krull Dimension for Sibirsky Graded Algebras
4.2 Hilbert Series for Sibirsky Graded Algebras S[sub(1)],m[sub(1)],m[sub(2)]....m[sub(l)],SI[sub(1)],m[sub(1)],m[sub(2)]....m[sub(l)]
4.3 Hilbert Series for Sibirsky Algebras S[sub(1,2)] , SI[sub(1,2)] and Their Krull Dimensions
4.4 Hilbert Series for Sibirsky Algebras S[sub(1,3)] , SI[sub(1,3)] and Their Krull Dimensions
4.5 Hilbert Series for Sibirsky Algebras S[sub(1,4)] , SI[sub(1,4)] and Their Krull Dimensions
4.6 Hilbert Series for Sibirsky Algebras S[sub(1,5)] , SI[sub(1,5)] and TheirKrull Dimensions
4.7 Obtaining Ordinary Hilbert Series for Sibirsky Algebras S[sub(1,2,3)]SI[sub(1,2,3)] Using, Residues, and Calculating Krull Dimensions for Them
4.8 Comments to Chapter Four
5 About the Center and Focus Problem
5.1 On a New Formulation of the Center and Focus Problem for Differential Systems s(1,m[sub(1)]m[sub(2)],m[sub(l)] )
5.2 Sibirsky Invariant Variety for Center and Focus
5.3 Focus Quantities L[sub(k)] and Constants G[sub(k)] on Sibirsky Invariant Variety of the System s(1,m[sub(1)]),...,m[sub(l)] and Null Focus Pseudo-Quantity
5.4 Polynomials in Coefficients of Differential Systems that Have Weight Isobarity (h,g)
5.5 Comments to Chapter Five
6 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the System s(1,m[sub(1)]),...,m[sub(2)]
6.1 Applications of Generating Functions and Hilbert Series to the Center and Focus Problem for the Differential System s(1,2)
6.2 Type of Generalized Focus Pseudo-Quantities for the Differential System s(1; 3)
6.3 On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities, that Take Part in Solving the Center and Focus Problem for the Differential System s(1; 3)
6.4 The Differential System s(1, 4) and Algebraically Independent Generalized Focus Pseudo-Quantities
6.5 On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities for the Differential System s(1,5)
6.6 Comitants that Have Generalized Focus Pseudo-Quantities of the Systems(1, 2, 3) as Coefficients, and Their Sibirsky Graded Algebra
6.7 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Differential System s(1,m[sub(1)],m[sub(l)])
1 ` 6.8 Comments to Chapter Six
7 On the Upper Bound of the Number of Functionally Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for Lyapunov System
7.1 Lie Operators of Representation of the Rotation Group SO(2,R) in the Space of Coefficients of Lyapunov System (5.6)
7.2 Comitants of System (7:1) (7:2) for the Rotation Group and Concept of Functional Basis
7.3 General Formulas that Interconnect Coefficients of Comitants of the Lyapunov System s(1,m[sub(1)] ,....,m[sub(l)] ) Among Themselves with Respect to the Rotation Group
7.4 On the Invariance of Focus Quantities in the Center and Focus Problem with Respect to the Rotation Group
7.5 On the Upper Bound of the Number of Algebraically Indepen-dent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Lyapunov System sL(1,m[sub(1)] ,....,m[sub(l)] )
7.6 Comments to Chapter Seven
Appendix 1
Appendix 2
Appendix 3
Appendix 4
Appendix 5
Appendix 6
Appendix 7
Appendix 8
Appendix 9
Appendix 10
Bibliography
Index

Citation preview

The Center and Focus Problem

Monographs and Research Notes in Mathematics Series Editors: John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky About the Series This series is designed to capture new developments and summarize what is known over the entire field of mathematics, both pure and applied. It will include a broad range of monographs and research notes on current and developing topics that will appeal to academics, graduate students and practitioners. Interdisciplinary books appealing not only to the mathematical community but also to engineers, physicists and computer scientists are encouraged. This series will maintain the highest editorial standards, publishing well-developed monographs and research notes on new topics that are final, but not yet refined into a formal monograph. The notes are meant to be a rapid means of publication for current material where the style of exposition reflects a developing topic. Applications of Homogenization Theory to the Study of Mineralized Tissue Robert P. Gilbert, Ana Vasilic, Sandra Klinge, Alex Panchenko, Klaus Hackl Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations Luca Lorenzi, Abdelaziz Rhandi Markov Random Flights Alexander D. Kolesnik Level-Crossing Problems and Inverse Gaussian Distributions Closed-Form Results and Approximations Vsevolod K. Malinovskii The Center and Focus Problem Algebraic Solutions and Hypotheses M.N. Popa & V.V. Pricop Abstract Calculus A Categorical Approach Francisco Javier Garcia-Pacheco For more information about this series please visit: https://www.crcpress.com/ Chapman–HallCRC-Monographs-and-Research-Notes-in-Mathematics/ bookseries/CRCMONRESNOT

The Center and Focus Problem Algebraic Solutions and Hypotheses

M.N. Popa & V.V. Pricop

First edition published 2022 by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 c 2022 M.N. Popa & V.V. Pricop  CRC Press is an imprint of Informa UK Limited The right of M.N. Popa & V.V. Pricop to be identified as authors of this work has been asserted by them in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-032-01725-9 (hbk) ISBN: 978-1-032-04410-1 (pbk) ISBN: 978-1-003-19307-4 (ebk) DOI: 10.1201/9781003193074 Typeset in Palatino by codeMantra

Contents

Authors Introduction 1 Lie Algebra of Operators of Centro-Affine Group Representation in the Coefficient Space of Polynomial Differential Systems 1.1 Two-dimensional Polynomial Differential Systems . . . . . . 1.1.1 Affine System . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 System with Quadratic Nonlinearities . . . . . . . . . 1.1.3 Quadratic System . . . . . . . . . . . . . . . . . . . . 1.1.4 System with Cubic Nonlinearities . . . . . . . . . . . . 1.1.5 Cubic System . . . . . . . . . . . . . . . . . . . . . . . 1.2 One-Parameter Linear Groups of Transformations of the Phase Plane of System (1.1)−(1.2) . . . . . . . . . . . . . . . . . . 1.3 Centro-Affine and Unimodular Transformation Groups of the Phase Plane of System (1.1)−(1.2) . . . . . . . . . . . . . . . 1.4 Lie Operators of One-Parameter Linear Groups and Their Representations in the Coefficient Space of System (1.1)−(1.2) 1.5 Operators of Representation of the Linear Groups (1.12), (1.14), (1.16) and (1.17) in the Space of Variables and Coefficients of System (1.1)−(1.2) . . . . . . . . . . . . . . . 1.6 Lie Algebra of Operators of Centro-Affine Group Representation in the Space of Variables and Coefficients of System (1.1)−(1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Comments to Chapter One . . . . . . . . . . . . . . . . . . . 2 Differential Equations for Centro-Affine Invariants and Comitants of Differential Systems and Their Applications 2.1 Concept of Centro-Affine Comitant and an Invariant of Differential System . . . . . . . . . . . . . . . . . . . . . . . 2.2 Centro-Affine Transformations of System (1.1)−(1.2) . . . . 2.3 Differential Equations for Centro-Affine Invariants and Comitants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Rational Absolute Centro-Affine Invariants and Comitants and Their Applications . . . . . . . . . . . . . . . . . . . . . . . .

ix 1

7 7 8 8 8 8 9 9 11 14

16

19 22

23 23 26 29 35 v

vi

Contents 2.5 2.6

Examples of Algebraic Bases of Centro-Affine Comitants and Invariants for Some Differential Systems . . . Comments to Chapter Two . . . . . . . . . . . . . . . . . . .

3 Generating Functions and Hilbert Series for Sibirsky Graded Algebras of Comitants and Invariants of Differential Systems 3.1 Formulas for Weights of Centro Affine Comitants and Invariants of Given Type . . . . . . . . . . . . . . . . . . . . 3.2 Initial form of Generating Function for Centro-Affine Comitants of Differential Systems . . . . . . . . . . . . . . . 3.3 Examples of Reduced Forms of Generating Functions for Centro-Affine Comitants of Differential Systems . . . . . 3.4 Hilbert Series for Graded Algebras of Unimodular Comitants and Invariants of Differential Systems . . . . . . . . . . . . . 3.5 Comments to Chapter Three . . . . . . . . . . . . . . . . . . 4 Hilbert Series for Sibirsky Algebras SΓ (SIΓ ) and Krull Dimension for Them 4.1 Krull Dimension for Sibirsky Graded Algebras . . . . . . . . 4.2 Hilbert Series for Sibirsky Graded Algebras S1,m1 ,m2 ,...,m` , SI1,m1 ,m2 ,...,m` . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hilbert Series for Sibirsky Algebras S1,2 , SI1,2 and Their Krull Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Hilbert Series for Sibirsky Algebras S1,3 , SI1,3 and Their Krull Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Hilbert Series for Sibirsky Algebras S1,4 , SI1,4 and Their Krull Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Hilbert Series for Sibirsky Algebras S1,5 , SI1,5 and Their Krull Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Obtaining Ordinary Hilbert Series for Sibirsky Algebras S1,2,3 , SI1,2,3 Using, Residues, and Calculating Krull Dimensions for Them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Comments to Chapter Four . . . . . . . . . . . . . . . . . . . 5 About the Center and Focus Problem 5.1 On a New Formulation of the Center and Focus Problem for Differential Systems s(1, m1 , m2 , ..., m` ) . . . . . . . . . . . . 5.2 Sibirsky Invariant Variety for Center and Focus . . . . . . . 5.3 Focus Quantities Lk and Constants Gk on Sibirsky Invariant Variety of the System s(1, m1 , ..., m` ) and Null Focus Pseudo-Quantity . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Polynomials in Coefficients of Differential Systems that Have Weight Isobarity (h, g) . . . . . . . . . . . . . . . . . . . . . 5.5 Comments to Chapter Five . . . . . . . . . . . . . . . . . . .

38 44

45 45 47 50 55 57

61 61 63 65 68 70 72

74 79 81 81 82

83 84 86

Contents 6 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the System s(1, m1 , ..., m ) 6.1 Applications of Generating Functions and Hilbert Series to the Center and Focus Problem for the Differential System s(1, 2) . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Type of Generalized Focus Pseudo-Quantities for the Differential System s(1, 3) . . . . . . . . . . . . . . . . . . 6.3 On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities, that Take Part in Solving the Center and Focus Problem for the Differential System s(1, 3) . . . . . . . . . . . . . . . . . . 6.4 The Differential System s(1, 4) and Algebraically Independent Generalized Focus Pseudo-Quantities . . . . 6.5 On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities for the Differential System s(1, 5) . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Comitants that Have Generalized Focus Pseudo-Quantities of the System s(1, 2, 3) as Coefficients, and Their Sibirsky Graded Algebra . . . . . . . . . . . . . . . . . . . . . . . 6.7 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Differential System s(1, m1 , ..., m ) . . . . . . . . . . . . . 6.8 Comments to Chapter Six . . . . . . . . . . . . . . . . .

vii

87

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87

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101

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109

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113

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127

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138

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144 147

7 On the Upper Bound of the Number of Functionally Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for Lyapunov System 149 7.1 Lie Operators of Representation of the Rotation Group SO(2, R) in the Space of Coefficients of Lyapunov System (5.6) . . . . . . . . . . . . . . . . . . . . . 149 7.2 Comitants of System (7.1) − (7.2) for the Rotation Group and Concept of Functional Basis . . . . . . . . . . . . . . . . 151 7.3 General Formulas that Interconnect Coefficients of Comitants of the Lyapunov System sL(1, m1 , ..., m ) Among Themselves with Respect to the Rotation Group . . . . . . . . . . . . . . 154 7.4 On the Invariance of Focus Quantities in the Center and Focus Problem with Respect to the Rotation Group . . . . . 156 7.5 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Lyapunov System sL(1, m1 , ..., m ) . . 162 7.6 Comments to Chapter Seven . . . . . . . . . . . . . . . . . . 163

viii

Contents

Appendix 1

165

Appendix 2

171

Appendix 3

175

Appendix 4

183

Appendix 5

187

Appendix 6

189

Appendix 7

191

Appendix 8

195

Appendix 9

197

Appendix 10

205

Bibliography

209

Index

213

Authors

Popa Mihail Nicolae was born on May 15, 1948, in V˘ alcinet¸ village, C˘ al˘ara¸si district, today Republic of Moldova. He graduated from the State University of Chisinau (today, the State University of Moldova) in 1971. Since 1975 he has been working at the Institute of Mathematics with Computing Center of the Academy of Sciences of Moldova (ASM) (today, Vladimir Andrunachievici Institute of Mathematics and Computer Science (IMCS)) in the Laboratory of Differential Equations. In the period 1980–1999, he worked as scientific secretary, 1999–2005 – deputy director, 2005–2010 – director of IMCS. From April 1, 2010, to until now, he is a senior scientific researcher at IMCS. In 1974–1979 he was a PhD student at ASM, and defended his doctoral thesis on “Affine classification of differential systems with quadratic nonlinearities” in 1979, at Gorky University (now Nizhny Novgorod, Russia). The doctor habilitation thesis entitled “Invariant processes in differential systems and their applications in qualitative theory” was defended in 1992 at the Institute of Mathematics of the National Academy of Ukraine, Kiev. Since 1996 he has been a tenured professor at the State University of Tiraspol (based in Chisinau), and in 2001 he was a visiting professor for a semester at the University of Limoges (France). He has trained 10 doctors of science. Mihail Popa’s scientific interests are related to the invariant processes in the qualitative theory of differential equations, Lie algebras and commutative graded algebras, generating functions and Hilbert series, orbit theory, Lyapunov stability theory. Pricop Victor Vasile was born on December 1, 1981, in Antone¸sti village, S¸tefan-Voda˘ district, today the Republic of Moldova. He graduated from the State University of Moldova in 2004. Since 2005 he has been working at the “Ion Creang˘ a” State Pedagogical University from Chi¸sin˘au. Since 2008 he has been working at Vladimir Andrunachievici Institute of Mathematics and Computer Science (IMCS) in the Laboratory of Differential Equations, currently scientific researcher. In 2009–2012 he was a PhD student at IMCS, and defended his doctoral thesis on “Combinatorial and asymptotic approaches based on graduated algebras and Hilbert series, applied to differential systems” in 2014, at IMCS. Since 2017 he has been a professor at the State Institute of International Relations of Moldova. Victor Pricop’s scientific interests are related to Lie algebras and graded algebras of invariants and comitants, generating functions and Hilbert series, applications of algebras to polynomial differential systems. ix

Introduction

One of the old problems of the qualitative theory of differential equations is the center and focus problem. It appears, for example, when the characteristic equation of system of differential equations dx dy = X(x, y), = Y (x, y) dt dt

(1)

has a singular point with purely imaginary eigenvalues (λ1,2 = ±i). This problem was formulated by French scientist H. Poincar´e (1854–1912) more than 130 years ago. It was shown that if a differential system cannot be solved explicitly, then it is possible to study the behavior of its solutions (integral curves) without knowledge of these solutions. In this way, the qualitative theory of differential equations was initiated. One of the most important problems of this theory is the study of behavior of integral curves (trajectories) around singular points, i.e. such points for which X(x, y) = Y (x, y) = 0. In this connection, Poincar´e proposed the following classification of nondegenerate singular points: saddle, node, center and focus. As it was noted above, the presence of center or focus at a singular point of differential system (1) is ensured by purely imaginary eigenvalues of the characteristic equation. Under this condition, in the case of a center, the singular point is surrounded by closed trajectories, and in the case of a focus, it is surrounded by spirals. The center and focus problem is to determine the condition under which a singular point is a center. In general, the center problem is algebraically unsolvable [2,19]. It should be noted that a large number of works in scientific centers of France, Russia, Belarus, China, Great Britain, Spain, Poland, Slovenia, Canada, the USA, etc. are dedicated to the center and focus problem and published in the world literature.

FIGURE 1 Singular points of the first type. DOI: 10.1201/9781003193074-1

1

2

Introduction

FIGURE 2 Singular points of the second type. In the Republic of Moldova, the first to deal with the center and focus problem for differential systems with polynomial nonlinearities was academician C. S. Sibirsky (1928–1990). His first work On conditions for the presence of a center and a focus (Kishinev Gos. Univ. Uch. Zap. 11, (1954), p. 115–117) caused interest to this problem in our country as well. His PhD thesis was focused on some aspects of the center and focus problem and was defended in 1955 at the Kazan University (Russia). At different stages, the disciples of the academician C. S. Sibirsky (N. I. Vulpe, A. S. S¸ub˘a, Iu. F. Calin, V. A. Baltag, D. V. Cozma and other) examined various issues of this problem and obtained important results. Later on, we will consider the case when the functions X(x, y) and Y (x, y) of differential system (1) are polynomials. The center and focus problem is algebraically solvable, if the right-hand sides of this system are nonzero. We write the considered system in the form `

`

dx X dy X = = Pmi (x, y), Qmi (x, y) (` < ∞), dt dt i=0 i=0

(2)

where Pmi and Qmi are homogeneous polynomials of degree mi ≥ 1 in x, y, and m0 = 1. The set {1, mi }`i=1 consists of a finite number (` < ∞) of distinct natural numbers. The coefficients and variables in polynomials of system (2) take values from the fields of real numbers R. Hereafter, we denote system (2) by s(m0 , m1 , ..., m` ). The fundamental results on the center and focus problem were obtained by A. M. Lyapunov (1857–1918) [20]. Henri Poincar´e and Aleksandr Lyapunov laid the foundations of the qualitative theory of differential systems. Using the methods proposed in [20,26], with the presence of singular point of the second group at the origin of coordinates O(0, 0), the condition of a center is that the infinite sequence of the following polynomials is equal to zero L1 , L2 , ..., Lk , ..., (3) whose variables are parameters of differential system (2), called focus quantities, Lyapunov’s constants or Poincar´e-Lyapunov’s constants.

Introduction

3

If at least one of quantities (3) is not zero, then coordinates origin O(0, 0) for system (2) is a focus. These conditions are necessary and sufficient. From Hilbert’s Theorem on the finiteness of basis of polynomial ideals, it follows that the essential center conditions, which imply vanishing of an infinite sequence of polynomials (3), consist of a finite number of polynomials, the rest ones are the consequences of them. Considering this result, the center and focus problem can be formulated in the following way: what finite number ω of polynomials from (3) (essential center conditions) Ln1 , Ln2 , ..., Lnω (ni ∈ {1, 2, ..., k, ...}; i = 1, ω, ω < ∞)

(4)

is necessary for their equality to zero annuls all polynomials from (3)? Hence, the center and focus problem consists of two parts. The first part relates to finding the number ω that determines the upper bound of the number of focus quantities which constitute the essential center conditions. The second part consists in finding the set Ω = {n1 , n2 , ..., nω } of indices ni (i = 1, ω), corresponding to the focus quantities, which constitute the essential center conditions. The generalized center and focus problem is to determine the upper bound of the number λ of algebraically independent focus quantities from Π = {Li : i ∈ Ω}. There is an opinion that if the center and focus problem is solved negatively for system s(1, m1 , ..., m` ), having at the origin a singular point of center or focus type, then the solution of the generalized center and focus problem can be considered as the final solution of this problem. The problem of determining essential center conditions (4) with number ω is a rather complicated problem and it is completely solved only for systems s(1, 2) and s(1, 3), for which we have ω = 3 and 5, respectively (see, e.g., [5], [23], [45]). Until now, it is not known the number ω for a system s(1, 2, 3), which seems to be not a complicated system. ˙ l¸adek, mostly based There exists a hypothesis formulated by Professor H. Zo on intuition, that for system s(1, 2, 3) the number ω is ≤ 13. Till now this hypothesis has not been disproved. But in [15], it was proved that the 12 focus quantities is not enough for solving the center and focus problem in the complex plane for system s(1, 2, 3). We note that initially some methods to solve the center and focus problem were proposed by Poincar´e and Lyapunov, who allowed obtaining solutions for systems s(1, 2) and s(1, 3) and other special cases. However, the specified way in solving the center and focus problem for system s(1, 2, 3) is connected with cumbersome computations with application of supercomputers. These difficulties are also insurmountable for other more complicated systems s(1, m1 , ..., m` ). Therefore, as a basis, it was taken the generalized center and focus problem, which was formulated above for any systems of the form s(1, m1 , ..., m` ). This

4

Introduction

allowed avoiding the calculation of the focus quantities (4) for given systems and replacing this process by investigating some Lie algebras of operators and Sibirsky graded algebras of comitants for the considered systems. To estimate the maximal number of algebraically independent focus quantities for system s(1, m1 , ..., m` ), these algebras were used. As a result, a finite upper bound for the number of algebraically independent focus quantities was obtained, which are involved in solving the center and focus problem for any system s(1, m1 , ..., m` ) from (1.1)–(1.2); this was announced for the first time at an international conference [31]. Results on the solution of the generalized center and focus problem were also presented at the international conferences on differential equations and algebra [30,32]. In addition, for the Lyapunov system sL(1, m1 , m2 , ..., m` ) from (7.9)–(7.10), it was found the upper bound of the number of functionally independent focus quantities, which are involved in solving of the center and focus problem for these systems. Chapter 1 (1.1–1.7) is devoted to the construction of Lie algebra of operators of representation of a centro-affine group in the space of coefficients and variables of differential systems with polynomial nonlinearities of the form (2). Chapter 2 (2.1–2.6) is dedicated to the investigation of differential equations for centro-affine invariants and comitants of system (2) and to the study of their algebraic bases. Chapter 3 (3.1–3.5) is devoted to the study of generating functions and Hilbert series for Sibirsky algebras of comitants and invariants of polynomial differential systems of the form (2). Chapter 4 (4.1–4.8) is dedicated to the construction of Hilbert series for Sibirsky algebras of different differential systems of the form (2) and to the computations of Krull dimension for these algebras. Chapter 5 (5.1–5.5) contains a brief summary of the concepts related to the new formulation of the center and focus problem for systems of the form (2). Chapter 6 (6.1–6.8) describes the examples of differential systems for which the upper bound of the number of algebraically independent focus quantities, which are involved in solving of the center and focus problem, is determined. These results are generalized for any system s(1, m1 , ..., m` ). Chapter 7 (7.1–7.6) is devoted to obtaining the upper bound of the number of functionally independent focus quantities, which are involved in solving the center and focus problem for the Lyapunov system sL(1, m1 , m2 , ..., m` ). This estimation is compared with the results established in Chapter 6. It is worth noting that the reader’s first acquaintance with the results of Chapter 7 will help him in better understanding the results of the first six chapters. The main result of this book can be concisely formulated as follows: Let ` P N = 2 (mi + 1) be the maximal possible number of nonzero coefficients of i=0

system (2), where m0 = 1. Then, the number of algebraically independent focus quantities from (3) does not exceed N − 1, which is the Krull dimension of

Introduction

5

Sibirsky algebra of comitants for system (2). It is also shown that this number can be reduced to N − 3, which is the Krull dimension of Sibirsky algebra of invariants for the mentioned system. It is assumed that the number of essential focus quantities ω from (4) does not exceed N − 1, and it can be improved up to N − 3; their construction will begin with the first algebraically independent nonzero focus quantities obtained consecutively up to the mentioned estimations. In this book, modern methods of algebra have found wide application. There were precisely these methods by which academician V. A. Andrunachievici (1917–1997) inspired the first author. The authors are extremely grateful to Professor N. I. Vulpe for useful discussions on the published articles [29], the main results of which are included in this monograph, and for advertising of the obtained results in many scientific centers of other countries. The authors are deeply grateful to the participants of the seminar of the Institute of Mathematics and Computer Science of ASM and the Tiraspol State University (Chisinau) “Differential Equations and Algebras”. The authors are sincerely grateful to the reviewers, Professors A. S. S ¸ ub˘ a and D. V. Cozma, for their critical comments and valuable advices in the elaboration of the present work. All disadvantages are on the authors’ conscience. Special thanks to Academician M. M. Ciobanu and journalist T. Rotaru for a popular presentation of the initial results of this book published in the article [9] to a wide range of readers.

1 Lie Algebra of Operators of Centro-Affine Group Representation in the Coefficient Space of Polynomial Differential Systems

1.1

Two-Dimensional Polynomial Differential Systems

Consider a two-dimensional autonomous polynomial system of differential equations x˙ =

` X

Pmi (x, y) ≡ P (x, y),

i=0

y˙ =

` X

(1.1) Qmi (x, y) ≡ Q(x, y) (` < ∞),

i=0 dy ` where x˙ = dx dt , y˙ = dt . We denote by Γ = {mi }i=0 some finite set of distinct nonnegative integers and homogeneities Pmi and Qmi of degree mi with respect to the phase variables x and y (i.e. Pmi (αx, αy) = αmi Pmi (x, y), Qmi
(αx, αy) = αmi Qmi (x, y) α ∈ R) on the right side of system (1.1) under mi i! = (mim −k)!k! given by equalities k

Pmi (x, y) = Qmi (x, y) =

mi  X mi  i 1 mi −k k ak x y , k

k=0 mi  X k=0

(1.2) mi  i 2 mi −k k ak x y , (mi ∈ Γ, i = 0, `). k

Note that all variables and coefficients of system (1.1)–(1.2) take values from the field of real numbers R. For the convenience, for the systems of the form (1.1)–(1.2), in some cases, we will use the notation s(m0 , m1 , ..., m` ) or s(Γ), where Γ = {mi }`i=0 , in which you can immediately see what degree of homogeneity is contained in the right parts of these systems. We give a simplified writing of some systems of the form (1.1)–(1.2), which will be needed in the future. DOI: 10.1201/9781003193074-2

7

8

1.1.1

The Center and Focus Problem

Affine System

If in (1.1)–(1.2), we take Γ = {0, 1}, then we obtain the differential system s(0, 1) x˙ = a + cx + dy, y˙ = b + ex + f y, (1.3) where

0

0

1

1

1

1

a = a10 , b = a20 , c = a10 , d = a11 , e = a20 , f = a21 .

(1.4)

We note that in system (1.3) a and b are called free members, cx + dy and ex + f y – linear parts.

1.1.2

System with Quadratic Nonlinearities

If in (1.1)–(1.2), we take Γ = {1, 2}, then we will obtain the differential system s(1, 2), which, in the simplified notation, accepted in many papers will have the form x˙ = cx + dy + gx2 + 2hxy + ky 2 , (1.5) y˙ = ex + f y + lx2 + 2mxy + ny 2 , where 0 0 0 0 1 1 1 c = a10 , d = a11 , e = a20 , f = a21 , g = a10 , h = a11 , k = a12 , (1.6) 1 1 1 l = a02 , m = a12 , n = a22 . We note that in the absence of linear part in system (1.5), we obtain the differential system s(2), which is called

1.1.3

Quadratic System

Which can be written as x˙ = gx2 + 2hxy + ky 2 ,

(1.7)

y˙ = lx2 + 2mxy + ny 2 .

1.1.4

System with Cubic Nonlinearities

If in (1.1)–(1.2), we take Γ = {1, 3}, then we will obtain the differential system s(1, 3), which, in the simplified notation, accepted in many papers and will have the form x˙ = cx + dy + px3 + 3qx2 y + 3rxy 2 + sy 3 ,

(1.8)

y˙ = ex + f y + tx3 + 3ux2 y + 3vxy 2 + wy 3 , where

0

0

0

0

1

1

1

1

1

1

1

1

c = a10 , d = a11 , e = a02 , f = a21 , p = a10 , q = a11 , r = a12 , s = a13 , t = a20 , u = a21 , v = a22 , w = a23 .

(1.9)

We note that in the absence of linear part in system (1.8), we obtain the differential system s(3), which is called

Lie Algebra of Operators of Centro-Affine Group

1.1.5

9

Cubic System

Which can be written as x˙ = px3 + 3qx2 y + 3rxy 2 + sy 3 , y˙ = tx3 + 3ux2 y + 3vxy 2 + wy 3 .

(1.10)

In the future, other systems of the form (1) will be considered with various Γ = {mi }i=0 , other than the abovementioned ones.

1.2

One-Parameter Linear Groups of Transformations of the Phase Plane of System (1.1)–(1.2)

Consider the transformations included in the one-parameter family {Tα }: x = f 1 (x, y, α), y = f 2 (x, y, α),

(1.11)

where α – a real parameter continuously changing in a certain range of R. To each value of the parameter α, there corresponds some transformation of family. Transformation (1.11) of the phase plane E 2 (x, y) means that the point (x, y) is transferred into new position (x, y) in the same plane E 2 (x, y). Definition 1.1. Let’s say that the family G1 = {Tα }, consisting of functions (1.11), continuously dependent on parameter α, forms a one-parameter transformation group G1 , if (1) Tα Tβ = Tγ , where Tγ ∈ {Tα } and γ = ϕ(α, β) are considered differentiable for the enough number of times; (2) Tα0 = I (or T0 = I ) (existence of a unity); (3) Tα−1 = Tα−1 (existence of the inverse element); (4) Tα (Tβ Tγ ) = (Tα Tβ )Tγ (associativity of multiplication in a group). Note that α−1 denotes the value of the parameter corresponding to the inverse transformation, and the condition (2) means the existence of a unique value of parameter α that guarantees the identity transformation in a group. Example 1.1. Consider the change of variables x = μx, y = y(μ ∈ R\{0}).

(1.12)

Take now two particular values μ and μ of parameter from R\{0} and successively apply substitution (1.12) and substitution Tμ x = μ x, y = y(μ ∈ R\{0}).

(1.13)

10

The Center and Focus Problem

Substituting (1.12) in these equalities, we obtain x = μμ x, y = y(μ, μ ∈ R\{0}). This shows that the result of applying two consecutive changes of variables (1.12) and (1.13) is identical to the result of applying the third transformation of this family with the value of the parameter μ = μμ . It is symbolically written as Tμ Tμ = Tμ , and they say that substitution (1.12) determines a group property. The existence of a unity in (1.12) is determined by the value of the parameter μ0 = 1. Backward substitution for (1.12) has the form x = μ−1 x, y = y, i.e. Tμ−1 = Tμ−1 and μ−1 = 1/μ. In this case, it is said that replacement (1.12) forms a transformation. To prove the associativity of transformation (1.12), another transformation Tμ is taken besides (1.13): x = μ x, y = y (μ ∈ R\{0}) and the property (4) of Definition 1.1 is directly checked. Therefore, the family {Tμ }, given by transformation (1.12), forms a group which we denote by M (2, R). Example 1.2. Consider the replacing of variables defined by a family {Tz }: x = x + zy, y = y (z ∈ R).

(1.14)

Similar to the previous Example 1.1, it can be shown that all the conditions of Definition 1.1 are performed, and family of substitutions (1.14) forms a group of continuous transformations from the parameter z, which we define through Z+ (2, R). It should be noted that in the case of transformation (1.14), the inverse transformation Tz−1 will consist of x = x − zy, y = y.

(1.15)

In this case, the value of the parameter corresponding to the inverse transformation will be z −1 = −z. In the future, we need two more one-parameter transformation groups, given by the following examples: Example 1.3. The group of transformation Z− (2, R), defined by the family of substitution {Th }: x = x, y = hx + y (h ∈ R).

(1.16)

Example 1.4. The group of transformation L(2, R), defined by the family of substitution {Tλ }: x = x, y = λy (λ ∈ R\{0}).

(1.17)

Remark 1.1. The groups M (2, R), Z+ (2, R), Z− (2, R), L(2, R) will be called linear one-parameter groups.

Lie Algebra of Operators of Centro-Affine Group

1.3

11

Centro-Affine and Unimodular Transformation Groups of the Phase Plane of System (1.1)–(1.2)

In this section, we give two linear groups that continuously depend on more than one parameter, the number of which is finite. If the number of parameters is r and there is no way to reduce it, the group is called r–parametric. The definition of r–parametric group coincides with Definition 2.1; only in this case, under the parameter α we will understand a certain vector with r–coordinates. Example 1.5. GL(2, R) – group of all centro-affine transformations of the phase plane E 2 (x, y): x = αx + βy, y = γx + δy,   α β = 0 , Δ = det γ δ

(1.18)

where x and y are new variables, and α, β, γ, δ ∈ R get continuously changing values. Check the execution of conditions of Definition 1.1 from family (1.18). If we consider the product of successive transformations (1.18) and x = α x + β  y, y = γ  x + δ  y,    α β  Δ = det = 0 , γ  δ

(1.19)

x = α (αx + βy) + β  (γx + δy) = α x + β  y, y = γ  (αx + βy) + δ  (γx + δy) = γ  x + δ  y,

(1.20)

given by equalities

then for their matrices, we have    β  αα + β  γ α =   αγ  + γδ  δ γ or which is the same 

α γ 

β  δ 



 =

α γ

β δ

α β + β  δ βγ  + δδ 



α γ

β δ

,

(1.21)

.

(1.22)

If we denote by Δ the determinant of matrix (1.21) for transformation (1.20), then from (1.22), we have Δ = Δ Δ, (1.23) and this determinant, according to (1.18) and (1.19), is nonzero. Therefore, transformation (1.20) is also a centro-affine transformation, i.e., the first condition of Definition 1.1 is satisfied.

12

The Center and Focus Problem

A transformation with the following parameters is taken as a unity of family (1.18) (identity centro-affine transformation): α = 1, β = 0, γ = 0, δ = 1.

(1.24)

With the help of (1.18), one can show the existence of the inverse element x= where

γ α δ β x + y, y = x + y, Δ Δ Δ Δ

(1.25)

β  = −β, γ  = −γ.

(1.26)

Similar to the first condition, one can check the associativity in the group. We note that the parameters α, β, γ, δ in (1.18) are independent since Δ=  0. Therefore, the number of parameters in this group cannot be reduced. This shows that the group GL(2, R) is a four-parameter group. Example 1.6. SL(2, R) – a group of all unimodular transformations of the phase plane E 2 (x, y): x = αx + βy, y = γx + δy,   α β Δ = det =1 , γ δ

(1.27)

where x and y are new variables, and α, β, γ, δ ∈ R get continuously changing values. We note that since Δ = 1, then the number of parameters can be reduced by one. Therefore, the unimodular group is a three-parameter group. Since all unimodular transformations are contained in the group of centroaffine transformations, we will say that they form a subgroup in the group of all centro-affine transformations. Remark 1.2. Each centro-affine transformation (1.18) can be considered as a composite of two transformations: 1

1

x = |Δ| 2 x, y = |Δ| 2 y and x=

α |Δ|

1 2

x+

β |Δ|

1 2

y, y =

γ |Δ|

1 2

x+

(1.28) δ 1

|Δ| 2

y,

(1.29)

where the second of them is unimodular. For proof of Remark 1.2, it is enough to calculate the product of successive transformations (1.28) and (1.29). We will consider transformations(1.18) or (1.27) to be given, if transforα β mation matrices are given by q = , and write q ∈ GL(2, R), or γ δ q ∈ SL(2, R). Following [1], we can show that there takes place

Lie Algebra of Operators of Centro-Affine Group

13

Theorem 1.1. Transformation (1.18), belonging to the four-parameter group GL(2, R), can be represented as a product of transformations (1.12), (1.14), (1.16) and (1.17), belonging, respectively, to one-parameter groups M (2, R), Z+ (2, R), Z− (2, R), L(2, R). Proof. Denote matrices corresponding to transformations (1.12), (1.14), (1.16) and (1.17), by 

μ 0

q1 = 

0 1

1 0 h 1

q3 =



 , q2 =



 , q4 =

1 0

z 1

1 0

0 λ

,

(1.30) ,

where μ, λ ∈ R\{0}. It is obvious that q1 , q2 , q3 , q4 ∈ GL(2, R). We note that performing transformations with matrices (1.30) in the following order q4 ((q1 q2 )q3 ) gives a transformation with matrix  q=

μ + μzh λh

μz λ

.

(1.31) 

In order that transformations (1.18) with the matrix q =

α γ

β δ

∈

GL(2, R) will be presented as (1.31), it is enough to take μ=

Δ βδ γ ,z= , h = , λ = δ (δ = 0). δ Δ δ

If δ = 0, then Δ = βγ = 0, and for this transformation, there can be written a matrix  α β q= , Δ 0 β which is equal to the product  q3 ·

α −αh +

Δ β

β −βh

.

According to the previous arguments, we find that the second factor in this product can be represented as q4 ((q1 q2 )q3 ), because in equality 

α −αh +

Δ β

β −βh



 =

α1 γ1

β1 δ1



the parameter δ1 = −βh is not zero, as in the case of arbitrary h β = 0, and it entails Δ = βγ = 0. Then, Theorem 1.1 is proven.

14

1.4

The Center and Focus Problem

Lie Operators of One-Parameter Linear Groups and Their Representations in the Coefficient Space of System (1.1)–(1.2)

Suppose that transformations (1.11) form a one-parameter linear group. It is clear that after transformation with elements of this group in system (1.1)– (1.2), this system does not change its form and can be written as follows: x˙ =

mi     mi  i 1 mi −k k bk x y , k i=0 k=0

mi     mi  i 2 mi −k k y˙ = bk x y . k i=0

(1.32)

k=0

i

Remark 1.3. We note that the coefficients bkj (j = 1, 2) of system (1.32) are linear functions of coefficients of system (1.1)–(1.2) with coefficients depending on the parameter α. The last statement can be written in the form i

i

bkj = g kj (A, α) (i = 0, ; j = 1, 2; k = 0, mi ),

(1.33)

where by A the set of coefficients of the right parts of system (1.1)–(1.2) is denoted. We note that equalities (1.33) define some group of linear transformations of the space of coefficients E N (A) of system (1.1)–(1.2), homomorphic with group (1.11) or, as they say, relations (1.33) define some linear representation of group (1.11) in the space E N (A). Similarly, a linear representation of the r–parametric linear group (1.11) is defined with α = (α1 , α2 , ..., αr ) on the space E N (A). By N we denote the number of coefficients of the right parts of system (1.1)–(1.2), which is defined by the equality

N =2

 

 mi +  + 1 .

(1.34)

i=0

Suppose that in group (1.11), identity transformation is provided by α = 0 (in the paper [24] it is proved that in any one-parameter group, you can take a new parameter with this condition, i.e., if Tα0 = I and α0 = 0, then with parameter redefinition, we can achieve T0 = I). We decompose functions (1.11) and (1.33) in a Taylor series by parameter α in neighborhood of α = 0. By condition T0 = I, we have f 1 (x, y, 0) = x, i

i

f 2 (x, y, 0) = y and g jk (A, 0) = akj (i = 0, ; j = 1, 2; k = 0, mi ). Therefore, by denoting

Lie Algebra of Operators of Centro-Affine Group i ∂g kj (A, α) ∂f j (x, y, α) ij , η k (A) = ξ (x, y) = ∂α ∂α α=0

15

j

α=0

(1.35)

(i = 0, ; j = 1, 2; k = 0, mi ), we write transformations (1.11) and (1.33) in the form x = x + ξ 1 (x, y)α + o(α), y = y + ξ 2 (x, y)α + o(α), i

i

i

bjk = akj + η kj (A)α + o(α) (i = 0, ; j = 1, 2; k = 0, mi ).

(1.36)

If we denote by B, a set of the coefficients of the right parts of systems (1.32), then, in this case, they say that groups (1.11)   and (1.33) are determined by i

their tangent vector field (ξ, η) = ξ 1 , ξ 2 , η jk

(i = 0, ; j = 1, 2; k = 0, mi ),

since by the formulas (1.35), the tangent vector is set at the point (x, y, A) to a curve described by the points (x, y, B) with group transformation (1.11) and (1.33). One-parameter groups (1.11) and (1.33) are fully recovered if the coordinates of the vector (ξ, η) are known. This process is carried out using the Lie equations with the initial condition [34]: dx = ξ 1 (x, y), x|α=0 = x, dα dy = ξ 2 (x, y), y |α=0 = y, dα dB = η(B), B|α=0 = A. dα

(1.37)

For any one-parameter group (1.11) and its representation (1.33), Lie equations (1.37) are written in a unique way and vice  versa.  i The tangent vector field (space) (ξ, η) = ξ 1 , ξ 2 , η jk (i = 0, ; j = 1, 2; k = 0, mi ) is also written using the first-order differential operator: X = ξ 1 (x, y) where

∂ ∂ + D, + ξ 2 (x, y) ∂x ∂y

mi   2   ∂ i D= η jk (A) i . ∂akj j=1 i=0 k=0 i

(1.38)

(1.39)

The functions ξ j (x, y) and η kj (A) are called coordinates of operators (1.38) and (1.39), and operator (1.39) is called operator of group representation (1.11) in the space of the coefficients E N (A) of system (1.1)–(1.2). The operator X (D) is called the infinitesimal Lie operator or simply Lie operator of a group of transformations (1.11) in the space E N +2 (x, y, A) (E N (A)), where N is from (1.34).

16

1.5

The Center and Focus Problem

Operators of Representation of the Linear Groups (1.12), (1.14), (1.16) and (1.17) in the Space of Variables and Coefficients of System (1.1)–(1.2)

Theorem 1.2. The operator of representation of the group M (2, R) from (1.12) in the space E N +2 (x, y, A) of system (1.1)–(1.2) has the form X1 = x

∂ + D1 , ∂x

(1.40)

where D1 =

mi ` X X

" (−mi + k +

i 1)a1k

i=0 k=0

∂ i

∂ a1k

+ (−mi +

i k)a2k

∂

#

i

.

(1.41)

∂ a2k

Proof. With transformation (1.12) in system (1.1)–(1.2), the coefficients of system (1.32) can be written as i

i

i

i

(1.42)

b1k = µ−mi +k+1 a1k , b2k = µ−mi +k a2k (i = 0, `, k = 0, mi ).

Note that for transformations (1.12) and (1.42) identity transformation is obtained with µ0 = 1. If we take |µ| = exp(µ) (µ ∈ R), then from (1.12) and (1.42) we have the following family of transformations {Tµ }: i

i

x = xexp(µ), y = y, b1k = a1k exp(−mi + k + 1)µ, i

b2k

=

i a2k exp(−mi

(1.43)

+ k)µ (i = 0, `, k = 0, mi ), µ ∈ R,

that satisfy the condition T0 = I, and herewith µ−1 = −µ. Since in (1.43), µ = 0 provides identity transformation in the space E N +2 (x, y, A), then taking into account (1.35) from (1.43), we have i

i

i

i

ξ 1 = x, ξ 2 = 0, η 1k = (−mi + k + 1)a1k , η 2k = (−mi + k)a2k . Substituting these equalities into (1.38)–(1.39) we find (1.40)–(1.41). Theorem 1.2 is proved. Consequence 1.1. The operator of representation of the group M (2, R) in the space of coefficients E N (A) of system (1.1)–(1.2) has the form (1.41). Theorem 1.3. The operator of representation of the group Z+ (2, R) from (1.14) in the space E N +2 (x, y, A) of system (1.1)–(1.2) has the form X2 = y

∂ + D2 , ∂x

(1.44)

Lie Algebra of Operators of Centro-Affine Group where

17



mi     ∂  ∂ i2 i1 i2 D2 = ak − kak−1 − k ak−1 i . i ∂ak1 ∂a2k i=0 k=0

(1.45)

Proof. If we make transformation (1.14) in system (1.1)–(1.2), then the coefficients of the resulting system (1.32) can be written as i

i

i

i

i

i

i

b1k = a1k + (a2k − ka1k−1 )z + o(z), bk2 = a2k − ka2k−1 z + o(z)

(1.46)

(i = 0, , k = 0, mi ), where o(z) is a polynomial containing z in the degree of at least two in all members. Using (1.14), (1.35) and (1.46), the coordinates of the tangent vector (ξ, η) of the group Z+ (2, R) are built, which have the form i

i

i

i

i

ξ 1 = y, ξ 2 = 0, η 1k = a2k − kak1 −1 , η k2 = −ka2k−1 . Substituting these equalities into (1.38)–(1.39), we find (1.44)–(1.45). Theorem 1.3 is proved. Consequence 1.2. The operator of representation of the group Z+ (2, R) in the space of coefficients E N (A) of system (1.1)–(1.2) has the form (1.45). Theorem 1.4. The operator of representation of the group Z− (2, R) from (1.16) in the space E N +2 (x, y, A) of system (1.1)–(1.2) has the form X3 = x where D3 =

mi   

(−mi +

i1 k)ak+1

i=0 k=0

∂ i

∂ak1

∂ + D3 , ∂y

+

i (ak1

(1.47)

+ (−mi +

∂ i k)a2k+1 ) i a2k

 .

(1.48)

Proof. If we make a transformation (1.16) in system (1.1)–(1.2), then the coefficients of the resulting system (1.32) can be written as i

i

i

b1k = a1k − (mi − k)a1k+1 h + o(h), i

b2k

=

i a2k

+

i [a1k

− (mi −

i k)a2k+1 ]h

(1.49)

+ o(h) (i = 0, , k = 0, mi ),

where o(h) is a polynomial containing h in the degree of at least two in all members. Using (1.16), (1.35) and (1.49), the coordinates of the tangent vector (ξ, η) of the Z− (2, R) are built, which have the form i

i

i

i

i

ξ 1 = 0, ξ 2 = x, η 1k = (−mi + k)a1k+1 , η 2k = a1k − (mi − k)a2k+1 .

18

The Center and Focus Problem

Substituting these equalities into (1.38)–(1.39), we find (1.47)–(1.48). Theorem 1.4 is proved. Consequence 1.3. The operator of representation of the group Z− (2, R) in the space of coefficients E N (A) of system (1.1)–(1.2) has the form (1.48). Theorem 1.5. The operator of representation of the group L(2, R) from (1.17) in the space E N +2 (x, y, A) of system (1.1)–(1.2) has the form X4 = y where D4 =

mi   

i −ka1k

i=0 k=0

∂ + D4 , ∂y

∂ i

∂ a1k

− (k −

(1.50)

i 1)a2k

∂

 .

i

∂ a2k

(1.51)

Proof. With transformation (1.17) in system (1.1)–(1.2), the coefficients of the resulting system (1.32) can be written as i

i

i

i

bk1 = λ−k ak1 , bk2 = λ−k+1 a2k (i = 0, ; k = 0, mi ).

(1.52)

We note that for transformations (1.17) and (1.52), the identity transformation is obtained by λ0 = 1. If |λ| = expλ, (λ ∈ R), then from (1.17) and (1.52) we have the following family of transformations {Tλ }: i

i

i

i

x = x, y = y expλ, b1k = a1k exp(−kλ), b2k = a2k exp(−k + 1)λ (i = 0, ; k = 0, mi ), λ ∈ R,

(1.53)

−1

that satisfies the condition T0 = I, and at the same time λ = −λ. Since in (1.53), λ = 0 provides identity transformation in the space E N +2 (x, y, A), then taking into account (1.35), from (1.53) we have i

i

i

i

ξ 1 = 0, ξ 2 = y, η 1k = −ka1k , η 2k = (−k + 1)a2k . Substituting these equalities into (1.38)–(1.39), we find (1.50)–(1.51). Theorem 1.5 is proved. Consequence 1.4. The operator of representation of the group L(2, R) from (1.17) in the space of coefficients E N (A) of system (1.1)–(1.2) has the form (1.51).

Lie Algebra of Operators of Centro-Affine Group

1.6

19

Lie Algebra of Operators of Centro-Affine Group Representation in the Space of Variables and Coefficients of System (1.1)–(1.2)

Definition 1.2. Following L. V. Ovsyannikov [24], let’s say that a linear space L on the field R is called Lie algebra, if for any two of its elements u, v the commutation operation [u, v] is defined, giving again an element L (commutator of elements u, v) and satisfying the following axioms: (1) bilinearity: for any u, v, w ∈ L and α, β ∈ R [αu + βv, w] = α[u, w] + β[v, w], [u, αv + βw] = α[u, v] + β[u, w]; (2) antisymmetry: for any u, v ∈ L [u, v] = −[v, u]; (3) Jacobi identity justice: for any u, v, w ∈ L [[u, v], w] + [[v, w], u] + [[w, u], v] = 0. The dimension of Lie algebra L is the dimension of its vector space L, and in the case of a finite dimension r, this algebra is denoted by the symbol Lr and is called finite-dimensional. In Section 1.5, a description of Lie operators of representation of one-parameter groups M (2, R), Z+ (2, R), Z− (2, R), L(2, R) in the spaces E N +2 (x, y, A) and E N (A) of system (1.1)–(1.2) was given, where N is from (1.34). These operators are the first-order differential operators whose general form can be written as follows: Y (F ) =

N +2 X

Pi

i=1

∂F , ∂yi

(1.54)

and a vector 0

0

`

(y1 , y2 , ..., yN +2 ) = (x, y, a10 , a11 , ..., a2m` ) ∈ E N +2 (x, y, A)

(1.55)

and Pi – are polynomials in y1 , y2 , ..., yN +2 . It is evident that Y (F1 + F2 ) = Y (F1 ) + Y (F2 ), Y (F1 F2 ) = F1 Y (F2 ) + F2 Y (F1 ), Y (α) = 0,

(1.56)

if F1 , F2 – are functions of y1 , y2 , ..., yN +2 , and α ∈ R. A composition Y1 Y2 of two differential operators of the form (1.54) is again a differential operator,

20

The Center and Focus Problem

but if Y1 and Y2 have order 1, then Y1 Y2 will have order 2, since it will already include the second derivatives. However, the fact that the commutator [Y1 , Y2 ] = Y1 Y2 − Y2 Y1

(1.57)

is again the first-order operator, follows from the fact that if for Y1 ,Y2 relations (1.56) are fulfilled, then they are fulfilled for [Y1 , Y2 ] as well. If we set operators Y1 and Y2 in a coordinate notation, Y1 =

N +2 

Pi

i=1

N +2  ∂ ∂ , Y2 = Qi , ∂y ∂yi i i=1

(1.58)

where Pi and Qi – are functions of y1 , y2 , ..., yN +2 , then taking into account (1.56) and (1.57), we have N +2 N +2    ∂Pi ∂ ∂Qi . (1.59) [Y1 , Y2 ] = Ri , Ri = Pk − Qk ∂yk ∂yi ∂yk i=1 i=1 This directly shows that [Y1 , Y2 ] is the first-order operator. This can be illustrated by: Example 1.7. Writing operators (1.40)–(1.41), (1.44)–(1.45), (1.47)–(1.48), (1.50)–(1.51) in the space E 8 (x, y, A) of system (1.3), where

we have

A = (a, b, c, d, e, f ),

(1.60)

∂ ∂ + D1 , X2 = y + D2 , ∂x ∂x ∂ ∂ X3 = x + D 3 , X4 = y + D4 , ∂y ∂y

(1.61)

∂ ∂ ∂ +d −e , ∂a ∂d ∂e ∂ ∂ ∂ ∂ + e + (f − c) −e , D2 = b ∂a ∂c ∂d ∂f ∂ ∂ ∂ ∂ D3 = a − d + (c − f ) +d , ∂b ∂c ∂e ∂f ∂ ∂ ∂ +e . D4 = b − d ∂b ∂d ∂e

(1.62)

X1 = x

and

D1 = a

Calculating all kinds of commutators of operators (1.61)–(1.62), we have [X1 , X2 ] = −X2 , [X1 , X3 ] = X3 , [X1 , X4 ] = 0, [X2 , X1 ] = X2 , [X2 , X3 ] = X4 − X1 , [X2 , X4 ] = −X2 , [X3 , X1 ] = −X3 , [X3 , X2 ] = X1 − X4 , [X3 , X4 ] = X3 , [X4 , X1 ] = 0, [X4 , X2 ] = X2 , [X4 , X3 ] = −X3 .

(1.63)

However, it should be noted that equality (1.57) has the following advantage:

Lie Algebra of Operators of Centro-Affine Group

21

Lemma 1.1. Commutator (1.57), whose operators are of the form (1.58), is invariant with respect to coordinate system (1.55) in E N +2 (x, y, A). The proof of Lemma 1.1 can be found in the paper of L. V. Ovsyannikov [24]. It is easy to check that the set of operators (1.54) form a linear space and satisfy Definition 1.2. Therefore, there takes place Lemma 1.2. Linear differential operators of the first-order (1.54) form a Lie algebra of operators. There is no associativity in this algebra, but it is replaced by a Jacobi identity. Suppose that dimension of a Lie algebra of linear differential operators is finite and is equal to r, and let us fix in this algebra some basis Y1 , Y2 , ..., Yr , which we denote by Lr . Consider commutators [Yμ , Yν ] of all possible pairs of these basis operators. Since any operator Y from Lr is decomposed over a basis as: r  Y = eμ Yμ , eμ = const, (1.64) μ=1

then the values of all [Yμ , Yν ] allow one to find uniquely the commutator of any operators from algebra Lr using bilinearity. From here it is clear that r–dimensional vector space Lr with the basis Y1 , Y2 , ..., Yr forms a Lie algebra if and only if, the commutators of basis operators belong to Lr , i.e. [Yμ , Yν ] =

r 

λ Cμν Yλ , (μ, ν = 1, r),

(1.65)

λ=1 λ – real numbers, called structural constants of the algebra Lr . A where Cμν convenient way to work directly with Lie algebra is the following: there is given the basis of operators and the table of their commutators, which is easily constructed using (1.56) and in which the value of [Yμ , Yν ] is located at the intersection of the μ-th row and the ν-th column. If commutators table consists of all zeros, then the Lie algebra is commutative. From here we have

Consequence 1.5. The one-dimensional Lie algebra of operators, i.e., Lie algebra consisting of one basis operator, is commutative. Remark 1.4. Using formulas (1.59), we can show that equalities (1.63) take place for operators (1.40)–(1.41), (1.44)–(1.45), (1.47)–(1.48), (1.50)–(1.51), applied to system (1.1)–(1.2) for any Γ = {mi }i=0 . From Remark 1.4, we consider the totality of these operators and draw up a Table 1.1 of their commutators, in which the value of [Yμ , Yν ] is located at the intersection of the μ-th row and the ν-th column. From here we note that the vector space with the basis (1.40), (1.44), (1.47) and (1.50) forms a 4-dimensional Lie algebra L4 .

22

The Center and Focus Problem X1 X2 X3 X4

X1 0 X2 −X3 0

X2 −X2 0 X1 − X4 X2

X3 X3 X4 − X1 0 − X3

X4 0 −X2 X3 0

Remark 1.5. If in Table 1.1, replace X1 , X2 , X3 , X4 with D1 , D2 , D3 , D4 from (1.41), (1.45), (1.48), (1.51), respectively, then, for new operators, this table will remain as it is, and, therefore, the vector space with this basis forms a 4-dimensional Lie algebra with Γ = {1}. Equalities (1.63) are called structural equations of a Lie algebra L4 . If we now consider that operators (1.40), (1.44), (1.47), (1.50) ((1.41), (1.45), (1.48), (1.51)) are operators of representations of one-parameter groups (1.12), (1.14), (1.16), (1.17) in the space E N +2 (x, y, A) (E N (A)), respectively, then according to Theorem 1.1, it should be noted that these one-parameter groups are the groups into which the group GL(2, R) (and its representation in the space E N +2 (x, y, A) (E N (A))) is decomposed. Thus, to representation of the group GL(2, R) in the space E N +2 (x, y, A) (E N (A)), the algebra Lie of operators X1 , X2 , X3 , X4 (D1 , D2 , D3 , D4 ) is put into correspondence, defined by Table 1.1 of commutators.

1.7

Comments to Chapter One

In this chapter, the continuous groups of linear transformations for twodimensional polynomial differential systems (1.1)–(1.2) are considered. It is shown that these transformations preserve the form of the systems. This leads us to the group theory and Lie algebras, without which it is impossible today to imagine modern mathematics and even physics. Therefore, the admission of these groups and the corresponding Lie algebras of operators to the considered systems implies the preservation of form of differential system (1.1)–(1.2) under the above-mentioned groups of linear transformations. As shown in [33,34], this fact is verified on the coordinates of the obtained operators X1 , X2 , X3 , X4 from (1.40)–(1.41), (1.44)–(1.45), (1.47)–(1.48), (1.50)–(1.51), respectively, taking into account that (1.38)– (1.39) satisfy the defining equations ξx1 P + ξy1 Q = ξ 1 Px + ξ 2 Py + D(P ), ξx2 P + ξy2 Q = ξ 1 Qx + ξ 2 Qy + D(Q),

(1.66)

of the respective system (1.1)–(1.2), where P and Q are from the considered system.

2 Differential Equations for Centro-Affine Invariants and Comitants of Differential Systems and Their Applications

2.1

Concept of Centro-Affine Comitant and an Invariant of Differential System

Recall that in  the future,  we will consider transformation (1.18) through the α β matrix q = ; and its membership in the group GL(2, R) we will γ δ write as q ∈ GL(2, R) and ∆ = det(q). Denote the set of coefficients of system (1.1)–(1.2) by A, and of (1.32) – by B. From Remark 1.3, it is clear that B = g(A, q), i.e., B is a linear function on A, and it is rational on elements of transformation q. Consider a few examples. Example 2.1. Consider a polynomial on the coefficients of system (1.3) and phase variables x, y, which we write in a determinant form   a x k1 (x, y, A) = det , (2.1) b y where we denote by A the set of coefficients of the right side of this system. Then, after transformation (1.18) in system (1.3), we obtain the system x˙ = a + c x + d y, y˙ = b + e x + f y,

(2.2)

where a = αa + βb, b = γa + δb, ∆c = δ(αc + βe) − γ(αd + βf ), ∆d = −β(αc + βe) + α(αd + βf ),

(2.3)

∆e = δ(γc + δe) − γ(γd + δf ), ∆f = −β(γc + δe) + α(γd + δf ). Similar expression (2.1) for system (2.2) will have a form   a x k1 (x, y, B) = det . b y DOI: 10.1201/9781003193074-3

(2.4) 23

24

The Center and Focus Problem

We will search which relation exists between (2.1) and (2.4). To do this, we note that the matrix form (2.4) will be written through (1.18) and (2.3) as   a x αa + βb αx + βy = , (2.5) γa + δb γx + δy b y where the right side will get the form   αa + βb αx + βy α = γa + δb γx + δy γ

β δ



Hence, taking into account (2.5), we have    a x α β a = γ δ b b y

x y

a b

x y

.

.

Applying a determinant property to this equality, we obtain    a x α β a x det = det det , γ δ b y b y from where, taking into account (1.18), (2.1), (2.4), we find the equality k1 (x, y, B) = Δk1 (x, y, A)

(2.6)

for any set of coefficients A of system (1.3), any x and y and any transformations q ∈ GL(2, R). Note also that from (2.6), the expression (2.4) is equal to the product of transformation determinant (1.18) on search expression (2.1). Example 2.2. Consider for system (1.3) matrix of the coefficients  c d F = e f

(2.7)

and write the sum of its elements along the main diagonal i1 (F ) = c + f.

(2.8)

It is usually denoted by trF and called as the trace of the matrix F . Then, for the respective system (2.2), received after transformation (1.18), in system (1.3), the similar expression has the form i1 (F ) = c + f .

(2.9)

i1 (F ) = i1 (F ),

(2.10)

Using equalities (2.3), we find

i.e., search expression (2.8) does not change its value after any transformation (1.18) in system (1.3).

Differential Equations for Comitants

25

Using (2.3), you can easily check that the sum of the elements of the second diagonal d + e of the matrix F from (2.7) does not have the above property. If we denote the set of coefficients of system (1.32), received after transformation (1.18) in system (1.1)–(1.2), by B, then Examples 2.1 and 2.2 lead us to the following Definition 2.1. We say that the integer rational function 0

0

0







2 K(x, y, a10 , a11 , ..., a1m1 , ..., a20 , a21 , ..., am ), 

which we will denote by K(x, y, A), on a set A of coefficients of system (1.1)– (1.2) and phase variables x and y, is called the centro-affine comitant of this system, if there exists such function λ(q), that the identity K(x, y, B) ≡ λ(q)K(x, y, A)

(2.11)

holds for any q ∈ GL(2, R), any coefficients of system (1.1)–(1.2) and any variables x and y. If the comitant K does not depend on variables x and y, then it is called the centro-affine invariant of system (1.1)–(1.2). Remark 2.1. In the monograph [37], it is shown that in (2.11), λ(q) = Δ−g , where g – integer number. The number g is usually called a weight of the comitant K(x, y, A). If g = 0, then the comitant K(x, y, A) is called absolute, otherwise – relative. In certain cases, in the name ”centro-affine comitant”, we will omit the word ”centro-affine”, if this does not lead to misunderstandings. From Examples 2.1 and 2.2, we obtain Observation 2.1. The expression k1 from (2.1) is a relative comitant with weight g = −1 for system (1.3), and i1 from (2.8) is an absolute invariant of system (1.3). Directly from Definition 2.1 follows Property 2.1. The product of any two centro-affine comitants (invariants) of system (1.1)–(1.2) is a centro-affine comitant (invariant) with a weight equal to the sum of weights of the factors. Property 2.2. The sum of two centro-affine comitants (invariants) of system (1.1)–(1.2) with the same weights is a centro-affine comitant (invariant) with the same weight. The sum of the two centro-affine comitants (invariants) is not always the centro-affine comitant (invariant). Observation 2.2. Similar to Examples 2.1 and 2.2, using (2.3), you can easily check that the following expressions are centro-affine invariants and

26

The Center and Focus Problem

comitants of system (1.3) with corresponding weights g i1 = c + f (g = 0), i2 = c2 + 2de + f 2 (g = 0), i3 = −ea2 + (c − f )ab + db2 , g = −1, k1 = −bx + ay (g = −1),

(2.12)

k2 = −ex2 + (c − f )xy + dy 2 (g = −1), k3 = −(ea + f b)x + (ca + db)y (g = −1). It is clear that, taking into account Properties 2.1 and 2.2, from (2.12), one can obtain an infinite number of centro-affine comitants and invariants of system (1.3).

2.2

Centro-Affine Transformations of System (1.1)–(1.2)

Lemma 2.1. The representation of the group of centro-affine transformations GL(2, R) with formulas (1.18) in the space of coefficients E N (A) of differential system (1.1)–(1.2) is a four-parameter group defined by one of series of expressions  k i ∂ ∂ mi 1 k (mi − k)! Δ bk = (−1) α +β [αPmi (δ, −γ) + βQmi (δ, −γ)], (mi )! ∂γ ∂δ (2.13)

 k i (mi − k)! ∂ ∂ γ α +β Pmi (δ, −γ) Δmi b2k = (−1)k ∂γ ∂δ (mi )!   k ∂ ∂ +δ α +β Qmi (δ, −γ) (i = 0, ; k = 0, mi ), (2.14) ∂γ ∂δ or i Δmi b1k

i

 mi − k k! ∂ ∂ = (−1) α γ +δ Pmi (−β, α) ∂α ∂β (mi )!   mi −k ∂ ∂ +β γ +δ Qmi (−β, α) , ∂α ∂β mi −k

Δmi b2k = (−1)mi −k

k! (mi )!

(2.15)

 mi −k ∂ ∂ γ +δ [γPmi (−β, α) ∂α ∂β

+ δQmi (−β, α)] (i = 0, , k = 0, mi ),

(2.16)

Differential Equations for Comitants

27

in which the value of the parameters α = δ = 1, β = γ = 0 corresponds to identity transformation (1.24). Proof. According to Remark 1.3, expressions (2.13), (2.14) or (2.15), (2.16) form a group of transformations of the coefficient space E N (A) of system (1.1)–(1.2). With centro-affine transformation (1.18) in system (1.1), taking into account (1.25)–(1.26), we find x˙ =



[αΔ−mi Pmi (δx + β  y, γ  x + αy)

mi ∈Γ

+ βΔ−mi Qmi (δx + β  y, γ  x + αy)],  y˙ = [γΔ−mi Pmi (δx + β  y, γ  x + αy) mi ∈Γ

+ δΔ−mi Qmi (δx + β  y, γ  x + αy)].

(2.17)

Taking into account (1.2) for Pmi and Qmi , we have Pmi (δx + β  y, γ  x + αy) =

mi   mi i 1 ak (δx + β  y)mi −k (γ  x + αy)k , k

k=0

Qmi (δx + β  y, γ  x + αy) =

mi   mi i 2 ak (δx + β  y)mi −k (γ  x + αy)k , k

k=0

or, which is the same mi   mi i 1 mi −k k Bk x y , k k=0 mi   mi i 2 mi −k k   Qmi (δx + β y, γ x + αy) = Bk x y , k Pmi (δx + β  y, γ  x + αy) =

(2.18)

k=0

i

i

where the coefficients B 1k and B 2k are rational functions of α, β  , γ  , δ i

and linear functions of akj . Then from (2.17), taking into account (2.18), we find mi    mi i 1 mi −k k x˙ = bk x y , k mi ∈Γ k=0 (2.19) mi    i m i 2 m −k k i y˙ = bk x y , k mi ∈Γ k=0

where i

i

i

i

i

i

Δmi b1k = αB 1k + βB 2k , Δmi b2k = γB k1 + δB 2k (i = 0, ).

(2.20)

28

The Center and Focus Problem Denote ξ = xy , η = xy , and from (2.18) we find mi   mi i 1 k Bk ξ , k k=0 mi   mi i 2 k   B k ξ (i = 0, ) Qmi (δ + β ξ, γ + αξ) = k Pmi (δ + β  ξ, γ  + αξ) =

(2.21)

k=0

and

mi   mi i 1 mi −k Pmi (δη + β , γ η + α) = Bk η , k k=0 mi   mi i 2 mi −k Bk η (i = 0, ). Qmi (δη + β  , γ  η + α) = k 



(2.22)

k=0

If we expand in a Taylor series polynomials (2.21) and (2.22) on variables ξ and η, respectively, then we obtain k mi k   ∂ ∂ ξ Pmi (δ + β  ξ, γ  + αξ) = α  + β Pmi (δ, γ  ), ∂δ k! ∂γ k=0 k mi k   ∂ ξ    ∂ Qmi (δ + β ξ, γ + αξ) = α  +β Qmi (δ, γ  ) ∂δ k! ∂γ k=0

(i = 0, )

(2.23)

and

 m i − k mi  η mi −k ∂ ∂ γ +δ  Pmi (β  , α), (mi − k)! ∂α ∂β k=0  mi −k mi  η mi −k ∂    ∂ γ +δ  Qmi (β  , α) Qmi (δη + β , γ η + α) = ∂α ∂β (mi − k)! Pmi (δη + β  , γ  η + α) =

k=0

(i = 0, ).

(2.24)

Equating coefficients by the same degrees ξ and η, respectively, in (2.21) and (2.23), as well as in (2.22) and (2.24), we obtain  k i (mi − k)! ∂ 1  ∂ Bk = α  +β Pmi (δ, γ  ), ∂γ ∂δ mi !  k i (mi − k )! ∂ ∂ B 2k = α  + β Qmi (δ, γ  ) (i = 0, ), ∂γ ∂δ mi ! or  mi −k i k! ∂ 1  ∂ Bk = γ +δ  Pmi (β  , α), ∂α ∂β mi !  mi −k i k! ∂ 2  ∂ Bk = γ +δ  Qmi (β  , α) (i = 0, ). ∂α ∂β mi !

Differential Equations for Comitants

29

From the last four equalities, taking into account (1.26), we have  k − k)! ∂ ∂ α +β Pmi (δ, −γ), = (−1) mi ! ∂γ ∂δ  k i (mi − k)! ∂ ∂ B k2 = (−1k ) α +β Qmi (δ, −γ) (i = 0, ), mi ! ∂γ ∂δ

(2.25)

 m i − k k! ∂ ∂ = (−1) γ +δ Pmi (−β, α), mi ! ∂α ∂β  mi −k i k! ∂ ∂ γ +δ Qmi (−β, α) (i = 0, ). B k2 = (−1)mi −k mi ! ∂α ∂β

(2.26)

k (mi

i

B 1k

or i

B 1k

mi −k

i

i

Taking into account (2.20) and (2.25)–(2.26), we obtain for b1k and b2k expressions (2.13)–(2.14) or (2.15)–(2.16). Lemma 2.1 is proved. Comparing (1.1)–(1.2) and (2.19), we obtain Consequence 2.1. On centro-affine transformation (1.18) system (1.1) with new coefficients and new variables retains its form, and its homogeneities of the right-hand sides with respect to x and y go to the homogeneities of the same degree with respect to x and y. From (2.13)–(2.14) and (2.15)–(2.16), we have i

i

Consequence 2.2. The expression Δmi b1k (Δmi b2k ) has the form (2.13) or (2.15) ((2.14) or (2.16)) and is a homogeneous function of degree k + 1 (k) with respect to the pair (α, β), and of degree mi − k (mi − k + 1) with respect to the pair (γ, δ). Consider expressions for the coefficients of affine differential system (1.3) after centro-affine transformation (1.18). Using Lemma 2.1 and equalities 0

0

1

(1.4), we obtain (2.3), where in system (2.2) we have a = b10 , b = b20 , c = b10 , 1

1

1

d = b11 , e = b20 , f = b21 .

2.3

Differential Equations for Centro-Affine Invariants and Comitants

Example 2.3. Consider Lie algebra of operators corresponding to the representation of the group GL(2, R) in the space E 6 (x, y, A) of system (1.3). According to Example 1.7, this Lie algebra consists of operators (1.61)–(1.62).

30

The Center and Focus Problem

For invariants and comitants (2.12) of system (1.3) on a group GL(2, R), using the mentioned operators, we obtain the equalities Dm (ij ) = 0 (m = 1, 4, j = 1, 2), D1 (i3 ) = D4 (i3 ) = i3 , D2 (i3 ) = D3 (i3 ) = 0 and X1 (kj ) = X4 (kj ) = i3 , X2 (kj ) = X3 (kj ) = 0 (j = 1, 3). The following theorem takes place Theorem 2.1. In order that an integer rational function of the coefficients of system (1.1)–(1.2) to be a centro-affine invariant of this system with a weight g, it is necessary and sufficient that it satisfies the equations D1 (I) = D4 (I) = −gI, D2 (I) = D3 (I) = 0,

(2.27)

where Dm (m = 1, 4) are from (1.41), (1.45), (1.48), (1.51) and form a Lie algebra of operators of representation of the group GL(2, R) in the space of coefficients E N (A) of system (1.1)–(1.2). Proof. Necessity. Suppose that I(A) is a centro-affine invariant of system (1.1)–(1.2) with the weight g. Then, according to Definition 2.1 and Remark 2.1, the following identity takes place I(B) = Δ−g I(A),

(2.28)

where the totality of B consists of the coefficients of system (2.19), having the form (2.13)–(2.16). Note that the determinant of transformation (1.18) satisfies the differential equations ∂Δ ∂Δ ∂Δ ∂Δ α +β = Δ, γ +δ = 0, ∂α ∂β ∂α ∂β (2.29) ∂Δ ∂Δ ∂Δ ∂Δ α +β = 0, γ +δ = Δ. ∂γ ∂δ ∂γ ∂δ If we apply to both sides of equality (2.28), the operators α

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +β , γ +δ , α +β , γ +δ , ∂α ∂β ∂α ∂β ∂γ ∂δ ∂γ ∂δ

then, taking into account equalities (2.29), we obtain ∂I(B) ∂I(B) ∂I(B) ∂I(B) +β = −gI(B), γ +δ = 0, ∂α ∂β ∂α ∂β ∂I(B) ∂I(B) ∂I(B) ∂I(B) α +β = 0, γ +δ = −gI(B). ∂γ ∂δ ∂γ ∂δ

α

(2.30)

This system of differential equations can be represented in another form, considering the fact that, according to equalities

Differential Equations for Comitants

31 0

0

(2.13)–(2.16), I(B) is a complex function of α, β, γ, δ included in b10 , b11 , ..., 0







b1m1 , ..., b20 , b21 , ..., b2m . Then, from (2.30), we obtain the following system of differential equations: mi   

⎡⎛

i

⎣ ⎝α

i=0 k=0

∂b1k

∂α ⎡⎛

i

+β

⎞

∂b1k ⎠

⎛ ∂I(B)

∂β

i

∂bk1

i

+ ⎝α

∂b2k

∂α ⎛

⎞

⎤

∂b2k ⎠

∂I(B) ⎦

i

+β

∂β

i

= −gI(B),

∂bk2

⎤ ⎞ ⎞ i i i i 1 1 2 2 b ∂b ∂I(B) ∂ ∂I(B) ∂b ∂b ⎣⎝γ k + δ k ⎠ ⎦ = 0, + ⎝γ k + δ k ⎠ i i ∂α ∂β ∂α ∂β 1 2 i=0 k=0 ∂ bk ∂bk ⎡⎛ i ⎞ ⎛ i ⎞ ⎤ i i mi   1 1 2 2  ⎣⎝α ∂bk + β ∂bk ⎠ ∂I(B) + ⎝α ∂bk + β ∂bk ⎠ ∂I(B) ⎦ = 0, i i ∂γ ∂δ ∂γ ∂δ i=0 k=0 ∂bk1 ∂b2k ⎡⎛ i ⎞ ⎛ i ⎞ ⎤ i i mi   1 1 2 2  ∂b ∂b B ∂I(B) ∂I( ) ∂b ∂b ⎣⎝γ k + δ k ⎠ ⎦ = −gI(B). + ⎝γ k + δ k ⎠ i i ∂γ ∂δ ∂γ ∂δ 1 2 i=0 k=0 ∂bk ∂bk (2.31) Consider the obtainment of simpler expressions for round brackets from (2.31). In view of equalities (2.29), we have mi   

⎞ i i i i 1 1 i ∂(Δmi b1k ) ∂(Δmi b1k ) ∂b ∂b mi ⎝ 1 mi k k⎠ Δ +β +β , α = −mi bk Δ + α ∂α ∂β ∂α ∂β ⎛

⎞ i i i i 2 2 i ∂(Δmi b2k ) ∂(Δmi b2k ) b ∂ ∂b mi ⎝ 2 mi k k⎠ Δ α +β = −mi bk Δ + α +β , ∂α ∂β ∂α ∂β

(2.32)

⎛

⎛ Δ

mi

⎝γ ⎛

Δ

mi

⎝γ

i

∂b1k ∂α

i

+δ

i

∂b2k ∂α

∂β i

+δ

⎞

∂b1k ⎠ ⎞

∂b2k ⎠ ∂β

i

i

∂(Δmi b1k ) ∂(Δmi b1k ) +δ , =γ ∂α ∂β i

(2.33)

(2.34)

i

∂(Δmi b2k ) ∂(Δmi b2k ) +δ , =γ ∂α ∂β

⎞ i i i i 1 1 ∂(Δmi b1k ) ∂b ∂(Δmi b1k ) ∂b mi ⎝ k k⎠ =α +β +β , α Δ ∂γ ∂δ ∂γ ∂δ

(2.35)

⎛

⎞ i i i i 2 2 ∂(Δmi b2k ) ∂b ∂(Δmi b2k ) ∂b mi ⎝ k k⎠ =α +β +β , α Δ ∂γ ∂δ ∂γ ∂δ

(2.36)

⎛

(2.37)

32

The Center and Focus Problem ⎛

⎞ i i i i 1 1 i ∂(Δmi b1k ) ∂b ∂b ∂ (Δmi b1k ) mi ⎝ 1 mi k k⎠ Δ = −mi bk Δ + γ +δ +δ , γ ∂γ ∂δ ∂γ ∂δ ⎞ i i i i 2 2 i ∂(Δmi b2k ) ∂b ∂b ∂ (Δmi bk2 ) mi ⎝ 2 mi k k⎠ Δ γ +δ . +δ = −mi bk Δ + γ ∂δ ∂γ ∂γ ∂δ

(2.38)

⎛

(2.39)

i

i

Considering the degree of homogeneity of the expressions Δmi b1k and Δmi b2k in relation to the couples (α, β) and (γ, δ), from Consequence 2.2, by Eulers theorem about homogeneity of the function for equalities (2.32)–(2.33) and (2.38)–(2.39), respectively, we obtain i

i

i ∂(Δmi b1k ) ∂(Δmi b1k ) α +β = (k + 1)Δmi b1k , ∂α ∂β i

i

i ∂(Δmi b2k ) ∂(Δmi b2k ) α +β = kΔmi b2k , ∂α ∂β i

i

(2.40)

i ∂(Δmi b1k ) ∂(Δmi b1k ) γ = (mi − k)Δmi b1k , +δ ∂γ ∂δ i

i

i ∂(Δmi b2k ) ∂(Δmi b2k ) γ +δ = (mi − k + 1)Δmi b2k . ∂γ ∂δ

Considering (2.15), for the expression from the left side of (2.34), we find ⎞ ⎛ i i  mi − k 1 1 (mi )! mi ⎝ ∂ bk ∂ ∂b ∂ (−1)mi −k +δ k⎠ = γ Δ γ +δ k! ∂α ∂β ∂α ∂β  mi −k+1 ∂ ∂ ×[γPmi (−β, α) + δQmi (−β, α)] + α γ +δ Pmi (−β, α) ∂α ∂β  mi −k+1 ∂ ∂ +β γ +δ Qmi (−β, α). ∂α ∂β From this equality, using (2.15) and (2.16), for k − 1, we have i

i

i i ∂b1 ∂b1 γ k + δ k = b2k − kb1k−1 . ∂α ∂β

(2.41)

For the left side of (2.35), considering (2.16), we obtain ⎞ ⎛ i i  mi −k+1 2 2 k! ∂ ∂ ∂b ∂b γ +δ Δmi ⎝γ k + δ k ⎠ = (−1)mi −k ∂α ∂β ∂α ∂β (mi )! i

× [γPmi (−β, α) + δQmi (−β, α)] = −kΔmi b2k−1 .

(2.42)

Differential Equations for Comitants

33

Using (2.13) for the left side of (2.36), we have ⎛

⎞ i i  k+1 1 1 ∂ ∂ ∂b ∂b mi ⎝ k (mi − k)! k⎠ k Δ α +β = (−1) α +β ∂γ ∂δ (mi )! ∂γ ∂δ i

× [αPmi (δ, −γ) + βQmi (δ, −γ)] = −(mi − k)Δmi b1k+1 .

(2.43)

Using (2.14) for the left side of (2.37), we have ⎛ (−1)k

i

i

∂b2k

⎞

∂ b2k ⎠

(mi )! Δmi ⎝α +β ∂γ ∂δ (mi − k)!

=

 k ∂ ∂ α +β ∂γ ∂δ

 k+1 ∂ ∂ ×[αPmi (δ, −γ) + βQmi (δ, −γ)] + γ α +β Pmi (δ, −γ) ∂γ ∂δ  k+1 ∂ ∂ +δ α +β Qmi (δ, −γ). ∂γ ∂δ Hence, considering (2.13) and (2.14) for k + 1, we have i

i

i i ∂b2 ∂b2 α k + β k = b1k − (mi − k)b2k+1 . ∂γ ∂δ

(2.44)

Then, using (2.40), (2.42) and (2.43) from (2.32)–(2.33), (2.35)–(2.36) and (2.38)–(2.39) after reduction by Δmi , we obtain i

i

i ∂b1 ∂b1 α k + β k = −(mi − k − 1)b1k , ∂α ∂β i

i

i ∂b2 ∂b2 α k + β k = −(mi − k)b2k , ∂α ∂β i

i

i ∂b2 ∂b2 γ k + δ k = −kb2k−1 , ∂α ∂β i

α

i

∂b1k

+β

∂γ

∂b1k ∂δ

i

i

= −(mi − k)b1k+1 , i

i ∂b1 ∂b1 γ k + δ k = −kb1k , ∂γ ∂δ i

i

i ∂b2 ∂b2 γ k + δ k = −(k − 1)b2k . ∂γ ∂δ

(2.45)

34

The Center and Focus Problem

Considering (2.41), (2.44) and (2.45), system (2.31) will be rewritten as mi   

⎡ i

1 ∂I(B)

⎣(mi − k − 1)bk

i=0 k=0 mi   

i

i

+ (mi − k)bk

i

= −gI(B),

∂b2k

∂bk1

⎡

⎤ ∂I(B) 2 ⎦

⎤

∂I(B) ∂I(B ) ∂I(B) 1 ⎣kbk−1 + kb2k−1 i − b2k i ⎦ = 0, i i=0 k=0 ∂b1k ∂b2k ∂b1k ⎡ ⎤ mi    i i i ∂I(B) ∂I(B) ∂I(B) 1 ⎣(mi − k)bk+1 + (mi − k)b2k+1 i − b1k i ⎦ = 0, i 1 i=0 k=0 ∂bk ∂b2 ∂b2k ⎡ ⎤ k mi    i i ⎣kb1k ∂I(B) + (k − 1)b2k ∂I(B) ⎦ = −gI(B). (2.46) i i i=0 k=0 ∂b1k ∂bk2 i

i

i

This system of equations must exist for any transformation (1.18). In particular, on the identity transformation α = δ = 1, β = γ = 0, using (1.1)–(1.2) i

i

and (1.18), we obtain the equalities bkj = akj (i = 0, , j = 1, 2, k = 0, mi ). Taking this into account, from (2.46) using Dm (m = 1, 4), from (1.41), (1.45), (1.48), (1.51) we find the necessity for conditions (2.27). Sufficiency. Suppose that an integer rational function I(A) satisfies equations (2.27) with operators (1.41), (1.45), (1.48), (1.51) for any A ∈ E N (A). Then, taking this into account, equalities (2.46) are valid and, therefore, equalities (2.30) are valid too. Note that if we formally apply L = ln|I(B)|, then from (2.30), we find the equalities ∂L ∂L ∂L ∂L α +β = −g, γ +δ = 0, ∂α ∂β ∂α ∂β (2.47) ∂L ∂L ∂L ∂L α +β = 0, γ +δ = −g. ∂γ ∂δ ∂γ ∂δ Considering (2.29), the function L = −gln|Δ| + ln|C|, where Δ is from (1.18), and C – arbitrary constant, satisfies these equalities as well. Therefore, from ln|I(B)| = −gln|Δ| + ln|C | we obtain I(B) = CΔ−g .

(2.48)

This also takes place on identity transformation α = δ = 1, β = γ = 0 i

i

of group (1.18) in system (1.1)–(1.2). Since in this case bkj = akj (i = 0, , j = 1, 2, k = 0, mi ), then C = I(A). Taking into account equality (2.48), we find identity (2.28), which, according to Definition 1.2, confirms the sufficiency of conditions (2.27).

Differential Equations for Comitants

35

For the fact that the operators Dm (m = 1, 4) form a Lie algebra of representation of the group GL(2, R) in the space E N (A), see Section 1.6. Theorem 2.1 is proved. Observation 2.3. The differential operators D2 and D3 contained in the equations of system (2.27) coincide with the operators Ω and Θ established in the paper [43]. Following the similar proof of Theorem 2.1, it can be shown that the following theorem takes place Theorem 2.2. In order that integer rational function K(x, y, A) to be a centro-affine comitant of system (1.1)–(1.2) with a weight g, it is necessary and sufficient that it satisfies the equations X1 (K) = X4 (K) = −gK, X2 (K) = X3 (K) = 0,

(2.49)

where Xm (m = 1, 4) are from (1.40), (1.44), (1.47), (1.50) and form a Lie algebra of operators of representation of the group GL(2, R) in the space E N +2 (x, y, A) of system (1.1)–(1.2).

2.4

Rational Absolute Centro-Affine Invariants and Comitants and Their Applications

Consider the case when in (2.11) K(x, y, A) is a fraction, i.e. its numerator and denominator are polynomials depending on the set A of coefficients of system (1.1)–(1.2) and on phase variables x, y. So, let R(x, y, A) K(x, y, A) = , (2.50) S(x, y, A) where R and S – reciprocal simple integer rational functions on the set A and phase variables x, y. If K(x, y, A) is a rational absolute centro-affine comitant, then from (2.11) will follow R(x, y, B)S(x, y, A) = R(x, y, A)S(x, y, B). (2.51) In equality (2.51), after substitution, instead of the set B, expressions from (2.13)–(2.14) or (2.15)–(2.16) and x, y from (1.18), the resulting ratio should turn into an identity with respect to the coefficients of system (1.1)–(1.2) and  α β  phase variables x, y for any value of α, β, γ, δ ∈ R, for which det = γ δ 0. Due to equality (2.51), the left side must be divided by R(x, y, A). However, R and S are reciprocal simple. Consequently, R(x, y, A) considered as a polynomial in coefficients of system (1.1)–(1.2) and phase variables x, y, should

36

The Center and Focus Problem

be divided by R(x, y, A). Since the degrees of these polynomials are obviously the same, then R(x, y, B) = λ1 (q)R(x, y, A). Similarly, for S(x, y, A) from (2.51) we find S(x, y, B) = λ2 (q)S(x, y, A). According to Remark 2.1, λ1 (q) = Δ−g1 and λ2 (q) = Δ−g2 , where g1 and g2 – integer numbers. It is not difficult to check that g1 = g2 . Consequently, R and S are relative centro-affine comitants with equal weights. The same can be said with respect to integer rational absolute centro-affine invariants. From here, we have that the following theorem takes place: Theorem 2.3. Every absolute rational centro-affine comitant (invariant) of system (1.1)–(1.2) is a quotient from dividing two integer rational centro-affine comitants (invariants) of this system with equal weights and vice versa. Observation 2.4. The idea of proving Theorem 2.3 is taken from paper [16]. Example 2.4. Consider expressions (2.12) which, according to Remark 2.1, are centro-affine invariants and comitants of system (1.3) with specific weights g. According to Theorem 2.3, one can easily check that, for example, the relations ii21 , ki31 , ki32 , ki33 , kk21 , kk31 , kk32 are absolute rational centro-affine invariants and comitants of system (1.3). From Theorems 2.1 and 2.2, it is easy to see that there takes place Lemma 2.2. In order that function (2.50) to be a rational absolute centroaffine comitant (invariant) of system (1.1)–(1.2), it is necessary and sufficient that it satisfies the equations Xi (K) = 0, (Di (K) = 0) (i = 1, 4),

(2.52)

where the expressions X1 , X2 , X3 , X4 (D1 , D2 , D3 , D4 ) from (1.40), (1.44), (1.47), (1.50) ((1.41), (1.45), (1.48), (1.51)) form a Lie algebra of operators. If you make a matrix M from the coordinates of the operators X1 , X2 , X3 , X4 , and matrix M1 – the same matrix, only for operators D1 , D2 , D3 , D4 , and denote their common ranks by R and R1 , respectively, then, using Lemma 2.2, from homogeneous linear partial differential equations (2.52), we obtain that there takes place Lemma 2.3. Maximal number of functionally independent rational absolute centro-affine comitants (invariants) of differential system (1.1)–(1.2) is equal to

 

     (2.53) 2 mi +  + 4 − R 2 m i +  + 2 − R1 . i=0

i=0

Differential Equations for Comitants

37

Note that in order that system (1.1)–(1.2) to possess m rational absolute centro-affine comitants and invariants, it is necessary that it has m + 1 centroaffine comitants and invariants from Definition 2.1. Taking into account this fact and Lemma 2.3, we obtain that for R = R1 = 4 there takes place Theorem 2.4. Maximal number of functionally independent rational absolute centro-affine comitants (invariants) of system (1.1)–(1.2) is equal to

=2

 

 mi + 

+1

 = 2

i=0

 

 mi + 

 − 1, Γ = {0}, {1} . (2.54)

i=0

In the formulation of Theorem 2.4, system (1.1)–(1.2) for Γ =  {0}, {1} for differential invariants is excluded from consideration, since, in the first case, there are no absolute invariants of the group GL(2, R), and in the second case, there are two of them. It is easy to show that in other cases, the matrix constructed on the coordinates of the operators X1 , X2 , X3 , X4 (D1 , D2 , D3 , D4 ), always has a nonzero fourth-order minor. Remark 2.2. Carrying out similar reasoning, as in the theory of invariants of binary forms (see, for example [1]), we can easily see that the indicated numbers in Theorem 2.4 are nothing more than the number of elements in the algebraic basis of comitants (invariants) of differential system (1.1)–(1.2), i.e., if we denote these centro-affine comitants by K1 , K2 ,...,K (invariants by I1 , I2 , ..., I ), then for any centro-affine comitant K (invariant I) it satisfies the equation P0 K m + P1 K m−1 + ... + Pm = 0 (Q0 I n + Q1 I n−1 + ... + Qn = 0),

(2.55)

where Pmi (i = 0, m) (Qi (i = 0, n)) are polynomials in K1 , K2 , ..., K (I1 , I2 , ..., I ). Therefore, in the future, the number  () from Theorem 2.4 we will consider to be related to the algebraic basis of centro-affine comitants (invariants) of system (1.1)–(1.2). Denote by S the coefficient at the highest degree xδ in the comitant K, which in the paper [43] is called semi-invariant. In the same monograph, it is shown that if S is a semi-invariant in the comitant K of system (1.1)–(1.2), then K = Sxδ − D3 (S)xδ−1 y +

1 2 (−1)δ δ D3 (S)xδ−2 y 2 + ... + D3 (S)y δ , 2! δ!

(2.56)

where D3 is taken from (1.48). Remark 2.3. Using equality (2.56), we can see that the comitants K1 , K2 , ..., K for system (1.1)–(1.2) are algebraically independent if and only if their semi-invariants are algebraically independent.

38

The Center and Focus Problem Suppose that comitant (2.56) is written as K = (S0 + α0 )xδ + (S1 + α1 )xδ−1 y + (S2 + α2 )xδ−2 y 2 + ... + (Sδ + αδ )y δ ,

(2.57)

where Si (i = 0, δ) – some known polynomials in coefficients of system (1.1)– (1.2), and αi (i = 0, δ) – unknown polynomials in coefficients of the same system. According to equality (2.56), for comitant (2.57), we obtain a system of partial differential equations D3 (S0 + α0 ) = −(S1 + α1 ), −D3 (S1 + α1 ) = S2 + α2 , D3 (S2 + α2 ) = −(S3 + α3 ), −D3 (S3 + α3 ) = S4 + α4 , ...................................................................................

(2.58)

(−1)k−1 D3 (Sk−1 + αk−1 ) = (−1)k (Sk + αk ) (k = 1, δ). Since the number of unknowns α0 , α1 , ..., αδ is greater than the number of equations (2.58) by one, then the following is true: Lemma 2.4. System (2.58) has an infinite set of solutions that generate infinite number of comitants of the form (2.57) for differential system (1.1)–(1.2).

2.5

Examples of Algebraic Bases of Centro-Affine Comitants and Invariants for Some Differential Systems

Example 2.5. For system (1.3), Lie algebra of operators of representation of a centro-affine group in the space E 8 (x, y, A) consists of operators (1.61)–(1.62). Considering this fact and system (1.3), writing the matrix of coordinates of these operators for it, we have ⎛ ⎞ x 0 a 0 0 d −e 0 ⎜ y 0 b 0 e f −c 0 −e ⎟ ⎟. M =⎜ (2.59) ⎝ 0 x 0 a −d 0 c−f d ⎠ 0 y 0 b 0 −d e 0 It can be easily checked that R = rank M = 4. Such conclusion was made based on the fact that, for example, for one of the fourth-order minors M , we have ⎛ ⎞ x 0 0 d ⎜ y 0 e f −c ⎟ ⎟ = −d k2 ≡ 0, det ⎜ ⎝ 0 x −d 0 ⎠ 0 y 0 −d

Differential Equations for Comitants

39

where k2 is from (2.12). Since in this case, for system (1.3), we have Γ = {0, 1}, i.e., it is obtained from system (1.1)–(1.2) for  = 1, m0 = 0, m1 = 1, then from (2.54) we find  = 5. Therefore, for system (1.3) an algebraic basis consists of five centro-affine invariants and comitants. We will show that as such, we can take the first five invariants and comitants from (2.12). For this, using all six invariants and comitants from (2.12), we construct the Jacobi matrix ⎛ ⎜ ⎜ ⎜ ⎜ J =⎜ ⎜ ⎜ ⎝

∂i1 ∂x ∂i2 ∂x ∂i3 ∂x ∂k1 ∂x ∂k2 ∂x ∂k3 ∂x

∂i1 ∂y ∂i2 ∂y ∂i3 ∂y ∂k1 ∂y ∂k2 ∂y ∂k3 ∂y

∂i1 ∂a ∂i2 ∂a ∂i3 ∂a ∂k1 ∂a ∂k2 ∂a ∂k3 ∂a

∂i1 ∂b ∂i2 ∂b ∂i3 ∂b ∂k1 ∂b ∂k2 ∂b ∂k3 ∂b

∂i1 ∂c ∂i2 ∂c ∂i3 ∂c ∂k1 ∂c ∂k2 ∂c ∂k3 ∂c

∂i1 ∂d ∂i2 ∂d ∂i3 ∂d ∂k1 ∂d ∂k2 ∂d ∂k3 ∂d

∂i1 ∂e ∂i2 ∂e ∂i3 ∂e ∂k1 ∂e ∂k2 ∂e ∂k3 ∂e

∂i1 ∂f ∂i2 ∂f ∂i3 ∂f ∂k1 ∂f ∂k2 ∂f ∂k3 ∂f

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(2.60)

Calculating all 28 minors of sixth order for this matrix, we find that all of them are equal to zero. Consequently, among the six centro-affine invariants and comitants (2.12) of system (1.3), there is an algebraic dependence that has the form (i1 k1 − k3 )2 − i2 k12 − 2i3 k2 + k32 = 0.

(2.61)

Note that in the theory of invariants of differential systems (see, for example [37]) a relation of the form (2.61) is called syzygy. Further, calculating with (2.12) the fifth-order minor (2.60), built on the lines 1,2,3,4 and 5 and columns 1,2,3,5 and 6, we obtain 2 3 2 2 2 2 2 2 2 Δ12356 12345 = 2[(−4a e + 4abce − 4abe f − b c e + 2b cef − b ef )x

+ (2a2 ce2 − 2a2 e2 f − abc2 e + 2abcef + 4abde2 − 2abef 2 − 2b2 cde + 2b2 def )y] ≡ 0. This allows us to conclude that the expressions i1 , i2 , i3 , k1 and k2 constitute an algebraic basis of centro-affine comitants of system (1.3). Remark 2.4. Similar to the previous case, it can be shown that any five centro-affine comitants and invariants from (2.12) form an algebraic basis of comitants for system (1.3). Remark 2.5. With the help of operators (1.62) and formulas for  from (2.54) it is easy to prove that the algebraic basis of invariants of system (1.3) consists of three elements. These may be i1 , i2 , i3 from (1.65). Example 2.6. Consider quadratic system of differential equations (1.7). With notation (1.6) and Lie algebras of operators of representation of centro-affine group in the space E 6 (A) of system (1.7) (see Section 1.5), we obtain that it

40

The Center and Focus Problem

consists of operators ∂ ∂ ∂ ∂ +k − 2l − m , ∂g ∂k ∂l ∂m ∂ ∂ ∂ ∂ ∂ D2 = l + (−g + m) + (−2h + n) −l − 2m , ∂g ∂h ∂k ∂m ∂n ∂ ∂ ∂ ∂ ∂ D3 = −2h −k + (−2m + g) + (−n + h) +k , ∂g ∂h ∂l ∂m ∂n ∂ ∂ ∂ ∂ − 2k +l −n . D4 = −h ∂h ∂k ∂l ∂n D1 = −g

(2.62)

Composing the matrix M1 , built on coordinates of the vectors of these operators, we obtain ⎛

−g 0 ⎜ l −g +m M1 = ⎜ ⎝ −2h −k −h 0

k −2h + n 0 −2k

−2l 0 −2m + g l

⎞ −m 0 −l −2m ⎟ ⎟. −n + h k ⎠ −n 0

(2.63)

It is easy to see that all 15 minors of the fourth order of this matrix are not identical to zero. For example, we give an expression of one of these minors, built on the columns 1, 2, 3, 4 of matrix (2.63), having a form Δ1234 = −2g 3 k + 2g 2 h2 − g 2 hn + 6g 2 km − 4gh2 m − 9ghkl −2ghmn + gkln − 4gkm2 + 8h3 l − 4h2 ln + 8hklm − 3k 2 l2 ≡ 0.

(2.64)

Consequently, for matrix (2.63), the total rank R1 = rankM1 = 4. Then, taking into account Remark 2.2 and Theorem 2.4 (in this case, we have  = 0, m0 = 2), we find  = 3, i.e., system (1.7) has an algebraic basis of centro-affine invariants consisting of three elements. We will search this basis among the invariants of this differential system, known from the paper [37] and having the form I7 = g 3 k − g 2 h2 + g 2 hn + g 2 km − 3gh2 m + 2ghkl + 2ghmn + 2gkln −gkm2 + gmn2 − h3 l − h2 ln − 4h2 m2 + 2hklm + hln2 − 3hm2 n +2klmn − km3 + ln3 − m2 n2 , I8 = g 3 k − g 2 h2 + g 2 hn − g 2 km − gh2 m + 4ghkl + 3gkm2 + gmn2 −3h3 l + 3h2 ln − 4h2 m2 − 4hklm − hln2 − hm2 n + 2k 2 l2 + 4klmn −3km3 + ln3 − m2 n2 ,

Differential Equations for Comitants

41

I9 = g 3 k − g 2 h2 + g 2 hn + 3g 2 km + 2g 2 n2 − 5gh2 m − 4ghmn +3gkm2 + gmn2 + h3 l + 3h2 ln − 4h2 m2 + 3hln2 − 5hm2 n +km3 + ln3 − m2 n2 , I15 = g 4 kn − g 3 h2 n − 2g 3 hkm + g 3 hn2 − g 3 k 2 l − g 3 kmn +2g 2 h3 m + 3g 2 h2 kl − 3g 2 h2 mn + 3g 2 hkln − 3g 2 k 2 lm − 3g 2 km2 n −g 2 mn3 − 2gh4 l − gh3 ln + 4gh3 m2 + 3gh2 klm + 3gh2 ln2 + 6ghkm3

(2.65)

+ghln3 + 3ghm2 n2 − 3gk 2 lm2 − 3gklmn2 + gkm3 n − gln4 + gm2 n3 −4h4 lm + h3 kl2 − 6h3 lmn + 3h2 kl2 n − 4h2 m3 n + 3hkl2 n2 −3hklm2 n + 4hkm4 + 2hlmn3 − 2hm3 n2 − k 2 lm3 + kl2 n3 −3klm2 n2 + 2km4 n. Applying Theorem 2.1 to these expressions, we obtain D1 (Ij ) = D4 (Ij ) = −2Ij , D2 (Ij ) = D3 (Ij ) = 0 (j = 7, 8, 9), D1 (I15 ) = D4 (I15 ) = −3I15 , D2 (I15 ) = D3 (I15 ) = 0, where Di (i = 1, 4) are from (2.62). This confirms that expressions (2.65) are centro-affine invariants of system (1.7). Composing Jacobi matrix for polynomials (2.65), we find ⎛ ⎜ ⎜ J =⎜ ⎝

∂I7 ∂g ∂I8 ∂g ∂I9 ∂g ∂I15 ∂g

∂I7 ∂h ∂I8 ∂h ∂I9 ∂h ∂I15 ∂h

∂I7 ∂k ∂I8 ∂k ∂I9 ∂k ∂I15 ∂k

∂I7 ∂l ∂I8 ∂l ∂I9 ∂l ∂I15 ∂l

∂I7 ∂m ∂I8 ∂m ∂I9 ∂m ∂I15 ∂m

∂I7 ∂n ∂I8 ∂n ∂I9 ∂n ∂I15 ∂n

⎞ ⎟ ⎟ ⎟. ⎠

(2.66)

Note that all 15 fourth-order minors of this matrix are zero. Therefore, among centro-affine invariants (2.65), there exists an algebraic relationship which has the form 2 (2.67) = 0. f1 ≡ (I8 − I7 )I92 + (I9 − I7 )I72 − 2I15 This syzygy is known from the paper [37]. Next, by calculating a constructed third-order minor, for example, with the first three lines and the first three columns of matrix (2.66), we find that it is nonzero. Consequently, the centroaffine invariants I7 , I8 , I9 can be taken as elements of the algebraic basis of the centro-affine invariants of system (1.7). It is not difficult to check that any three invariants of (2.65) also form an algebraic basis of the centro-affine invariants of system (1.7), i.e.,  = 3. Example 2.7. Consider the cubic system of differential equations (1.10). Using notation (1.9) and Lie algebra of operators of representation of centroaffine group in the space E 8 (A) of system (1.10) (see Section 1.5), we obtain that algebra consists of operators

42

The Center and Focus Problem ∂ ∂ ∂ ∂ ∂ ∂ −q +s − 3t − 2u −v , ∂p ∂q ∂s ∂t ∂u ∂v ∂ ∂ ∂ ∂ D2 = t + (−p + u) + (−2q + v) + (−3r + w) ∂p ∂q ∂r ∂s ∂ ∂ ∂ −t − 2u − 3v , ∂u ∂v ∂w ∂ ∂ ∂ ∂ ∂ D3 = −3q − 2r −s + (−3u + p) + (−2v + q) ∂p ∂q ∂r ∂t ∂u ∂ ∂ + (−w + r) +s , ∂v ∂w ∂ ∂ ∂ ∂ ∂ ∂ D4 = −q − 2r − 3s +t −v − 2w . ∂q ∂r ∂s ∂t ∂v ∂w

D1 = −2p

(2.68)

Composing the matrix M1 , built on coordinates of the vectors of these operators, we obtain

M1 =

−2p t −3q 0

−q −p + u −2r −q

0 −2q + v −s −2r

s −3r + w 0 −3s

−3t 0 −3u + p t

−2u −t −2v + q 0

−v −2u −w + r −v

0 −3v s −2w

 .

(2.69) It is easy to verify that all 70 fourth-order minors of this matrix are not identically equal to zero. For example, we give an expression of one of such minors, built on the columns 1,2,3 and 4 of matrix (2.69), having the form Δ1234 = 6p2 s2 − 36pqrs + 2pqsw + 24pr3 − 8pr2 w + 12prsv −6ps2 u + 24q 3 s − 18q 2 r2 + 6q 2 rw − 12q 2 sv + 6qrsu + 4qs2 t − 4r2 st.

(2.70)

Consequently, for matrix (2.69), the total rank R1 = rankM1 = 4. Then, considering Remark 2.2 and Theorem 2.4 (we have  = 0, m0 = 3 in it), we find  = 5, i.e., system (1.10) has an algebraic basis of centro-affine invariants consisting of five elements. We will search for this basis among the invariants of this differential system, known from the paper [43] and having a form J1 = 2pr + 2pw − 2q 2 − 4qv + 2ru + 2uw − 2v 2 , J2 = 2pr − 2q 2 + 2qv − 4ru + 2st + 2uw2 − 2v 2 , J3 = −p2 s + 3pqr − pqw + 5prv − 2psu + pvw − 2q 3 − 2q 2 v − qru − 5quw +2qv 2 + r2 t + 2rtw + ruv − su2 + tw2 − 3uvw + 2v 3 , J4 = −p2 s + 3pqr − pqw + 2prv + psu + pvw − 2q 3 + q 2 v − 4qru − 3qst −2quw − qv 2 + 4r2 t − rtw + 4ruv + 3stv − 4su2 + tw2 − 3uvw + 2v 3 , J5 = −p2 rw + p2 sv + pq 2 w − 2pqsu + pr2 u − 2pruw + 2prv 2 − puw2

Differential Equations for Comitants

43

+pv 2 w − q 3 v + q 2 ru + q 2 st + 2q 2 uw − 2q 2 v 2 − qr2 t + 2qstv − 2qsu2 +qtw2 − qv 3 − 2r2 tv + 2r2 u2 − 2rtvw + ru2 w + ruv 2 + stv 2 − su2 v, J6 = 2p3 w3 − 18p2 qvw2 + 6p2 ruw2 + 12p2 rv 2 w − 6p2 stw2 + 12p2 suvw −12p2 sv 3 + 12pq 2 uw2 + 42pq 2 v 2 w + 12pqrtw2 − 120pqruvw + 24pqstvw −24pqsu2 w + 36pqsuv 2 − 24pr2 tvw + 78pr2 u2 w − 24prstuw +24prstv 2 − 36prsu2 v + 6ps2 t2 w − 12ps2 tuv + 12ps2 u3 − 12q 3 tw2 −42q 3 v 3 + 36q 2 rtvw + 126q 2 ruv 2 + 24q 2 stuw − 78q 2 stv 2 − 36qr2 tuw −126qr2 u2 v − 12qrst2 w + 120qrstuv − 6qs2 t2 v − 12qs2 tu2 +12r3 t2 w + 42r3 u3 − 12r2 st2 v − 42r2 stu2 + 18rs2 t2 u − 2s3 t3 . (2.71) Applying Theorem 2.1 to these expressions, we obtain D1 (Jj ) = D4 (Jj ) = −2Jj , D2 (Jj ) = D3 (Jj ) = 0 (j = 1, 2), D1 (Jj ) = D4 (Jj ) = −3Jj , D2 (Jj ) = D3 (Jj ) = 0 (j = 3, 4), D1 (J5 ) = D4 (J5 ) = −4J5 , D2 (J5 ) = D3 (J5 ) = 0, D1 (J6 ) = D4 (J6 ) = −6J6 , D2 (J6 ) = D3 (J6 ) = 0, where Di (i = 1, 4) are from (2.68). This confirms that expressions (2.71) are centro-affine invariants of system (1.10). Composing the Jacobi matrix for the polynomials (2.71), we have ⎛ ∂J1 ∂J1 ∂J1 ∂J1 ∂J1 ∂J1 ∂J1 ∂J1 ⎞ ⎜ ⎜ ⎜ ⎜ J =⎜ ⎜ ⎜ ⎝

∂p ∂J2 ∂p ∂J3 ∂p ∂J4 ∂p ∂J5 ∂p ∂J6 ∂p

∂q ∂J2 ∂q ∂J3 ∂q ∂J4 ∂q ∂J5 ∂q ∂J6 ∂q

∂r ∂J2 ∂r ∂J3 ∂r ∂J4 ∂r ∂J5 ∂r ∂J6 ∂r

∂s ∂J2 ∂s ∂J3 ∂s ∂J4 ∂s ∂J5 ∂s ∂J6 ∂s

∂t ∂J2 ∂t ∂J3 ∂t ∂J4 ∂t ∂J5 ∂t ∂J6 ∂t

∂u ∂J2 ∂u ∂J3 ∂u ∂J4 ∂u ∂J5 ∂u ∂J6 ∂u

∂v ∂J2 ∂v ∂J3 ∂v ∂J4 ∂v ∂J5 ∂v ∂J6 ∂v

∂w ∂J2 ∂w ∂J3 ∂w ∂J4 ∂w ∂J5 ∂w ∂J6 ∂w

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(2.72)

Note that all 28 minors of the sixth order of this matrix are zero. Therefore, among centro-affine invariants (2.71), there exists an algebraic relationship (syzygy), which has the form ϕ1 ≡ 18J16 − 81J15 J2 + 189J14 J22 + 108J14 J5 − 279J13 J23 − 108J13 J2 J5 −30J13 J32 + 96J13 J3 J4 − 12J13 J42 − 144J13 J6 + 243J12 J24 − 324J12 J22 J5 +162J12 J2 J32 − 432J12 J2 J3 J4 + 108J12 J2 J42 + 324J12 J2 J6 − 108J1 J25 +540J1 J23 J5 − 196J1 J22 J32 + 504J1 J22 J3 J4 − 144J1 J22 J42 − 432J1 J22 J6 −648J1 J2 J52 + 288J1 J32 J5 − 144J1 J3 J4 J5 − 144J1 J42 J5 − 432J1 J5 J6 +18J26 − 216J24 J5 + 66J23 J32 − 168J23 J3 J4 + 48J23 J42 + 144J23 J6 +648J22 J52 − 720J2 J32 J5 + 1008J2 J3 J4 J5 − 288J2 J42 J5 − 864J2 J5 J6 +128J34 − 416J33 J4 + 480J32 J42 + 264J32 J6 − 224J3 J43 − 672J3 J4 J6 +32J44 + 192J42 J6 − 2592J53 + 288J62 = 0.

(2.73)

44

The Center and Focus Problem

Next, by calculating the constructed fifth-order minor, for example, with the first five lines and the first five columns of matrix (2.72), we find that it is nonzero. Consequently, the centro-affine invariants J1 − J5 can be taken as elements of the algebraic basis of the centro-affine invariants of system (1.10). It is not difficult to check that any five invariants from (2.71) also form an algebraic basis of the centro-affine invariants of system (1.10).

2.6

Comments to Chapter Two

Centro-affine comitants and invariants of differential systems of the form (1.1)– (1.2) have found wide applications in qualitative study of such systems (see, e.g., [34], [37], [43]). However, the existing methods of their construction [37,43] do not allow us to know a priori the number of invariants and comitants in algebraic basis and other bases, the number of which differs from system to system. In this chapter, general formulas for the number of invariants and comitants included in an algebraic basis of any system of the form (1.1)–(1.2) are given.

3 Generating Functions and Hilbert Series for Sibirsky Graded Algebras of Comitants and Invariants of Differential Systems

3.1

Formulas for Weights of Centro Affine Comitants and Invariants of Given Type

Note that the index i over the coefficients of system (1.1)–(1.2) indicates i

that ajk belong to homogeneities Pmi (x, y) and Qmi (x, y), and ` indicates the number of homogeneities in the right side of system (1.2). Definition 3.1. Let’s say that centro-affine comitant of system (1.1)–(1.2) has type (d) = (δ, d0 , d1 , ..., d` ), (3.1) if it is a homogeneous polynomial of degree di with respect to the coefficients i ajk of homogeneities of Pmi (x, y) and Qmi (x, y) and of degree δ with respect P` to the phase variables x, y. At the same time, the number d = i=0 di (δ) is called the degree (order) of comitant of type (3.1). Observation 3.1. From Definition 2.1 of centro-affine invariant as a comitant, in which the phase variables x and y are absent, it follows that for an invariant of system (1.1)–(1.2) of type (3.1) in this record δ = 0. If we consider, for example, type (3.1) for centro-affine comitants of system (1.3), then it will be written as (d) = (δ, d0 , d1 ),

(3.2)

since on the right side of this system, there are two homogeneities: of zero degree and the linear one. Example 3.1. Consider the types of centro-affine invariants of system (1.3), given in (2.12) from (3.2): (1) invariant i1 has a type (0, 0, 1); (2) invariant i2 has a type (0, 0, 2); DOI: 10.1201/9781003193074-4

45

46

The Center and Focus Problem (3) (4) (5) (6)

invariant comitant comitant comitant

i3 has a type (0, 2, 1); k1 has a type (1, 1, 0); k2 has a type (2, 0, 1); k3 has a type (1, 1, 1).

Example 3.2. The centro-affine invariants I7 , I8 , I9 and I15 from (2.65) are invariants of type (0, 4) and (0, 6) for system (1.7). If we consider centro-affine invariants (2.71) for system (1.10), then J1 , J2 are invariants of type (0, 2), J3 , J4 – of type (0, 3), J5 – of type (0, 4), J6 – of type (0, 6). Lemma 3.1. If the centro-affine comitant K(x, y, A) of system (1.1)–(1.2) has type (3.1) and weight g, then for Γ = {mi }i=0 , the following equality is true:   (3.3) 2g = di (mi − 1) − δ. i=0

Proof. Let there be given a centro-affine comitant K(x, y, A) in the form 

K(x, y, A) =

 

C

i

i

i

i

i

i

i

i

i

i

i

i

(a10 )η0 (a11 )η1 ...(a1mi )ηmi (a20 )ξ0 (a21 )ξ1 ...(a2mi )ξmi

i=0

(3.4)

×xδ1 y δ2 , having type (3.1) and weight g, where C – numerical coefficients. Therefore, from (3.4), according to Definition 3.1, we have δ1 + δ2 = δ, i

i

i

i

i

(3.5)

i

i

η 0 + η 1 + ... + η mi + ξ 0 + ξ 1 + ξ 2 + ... + ξ mi = di , (i = 0, ).

(3.6)

By implementing the following centro-affine transformation in system, (1.1)–(1.2) (3.7) x = μ−1 x, y = μ−1 y, Δ = μ−2 , μ ∈ R\{0}, we obtain system (2.19), in which the new coefficients are of the form i

i

bkj = μmi −1 akj (j = 1, 2, i = 0, , k = 0, mi ).

(3.8)

On the other hand, if we use the fact that K(x, y, A) is a centro-affine comitant of system (1.1)–(1.2) of the weight g, then according to Definition 2.1 and Remark 2.1 for transformation (3.7) with the help of (3.4), we have 

C

 

i

i

i

i

i

i

i

i

i

i

i

i

1 (b10 )η0 (b11 )η1 ...(bm )ηmi (b20 )ξ0 (b21 )ξ1 ...(b2mi )ξmi xδ1 y δ2 i

i=0

= μ2g



C

  i=0

i

i

i

i

i

i

i

i

i

i

i

i

1 (a10 )η0 (a11 )η1 ...(am )ηmi (a02 )ξ0 (a21 )ξ1 ...(a2mi )ξmi xδ1 y δ2 . i

Generating Functions and Hilbert Series

47

Substituting (3.7) and (3.8) in the left side of this equality, we find μ



i i i i i i i=0 (mi −1)(η 0 +η 1 +...+η mi +ξ 0 +ξ 1 +...+ξ mi )−(δ1 +δ2 )

= μ2g ,

from where, taking into account (3.5) and (3.6), we obtain equality (3.3). Lemma 3.1 is proved. Consequence 3.1. Centro-affine comitants or invariants of system (1.1)– (1.2), having the same type (3.1), have the same weight. The proof of Consequence 3.1 follows directly from equality (3.3). Observation 3.2. Equality (3.3) for centro-affine comitants of system (1.1)– (1.2) are known from the paper [43].

3.2

Initial form of Generating Function for Centro-Affine Comitants of Differential Systems

Lemma 3.2. The set of centro-affine comitants of system (1.1)–(1.2) of the same type (3.1) forms a finite-dimensional linear space, i.e. has a finite maximal system of linearly independent comitants (linear basis) of given type through which all others are linearly expressed. Proof. Let there exist the centro-affine comitants of system (1.1)–(1.2) of type (3.1). Then, according to Property 2.2 and Consequence 3.1, for any two such comitants K and k of this set and any numbers α, β ∈ R we have K + k, and αK belongs to this set, and there are equalities α(K + k) = = αK + αk, (α + β)K = αK + βK, (αβ)K = α(βK), 1 · K = K. For any comitant K, there is an inverse element (−K). The sum of comitants is commutative and associative, and the zero element is considered to be a comitant that is identically equal to zero, which can be considered a comitant of any weight and any type. Therefore, this set forms a linear space. From the monographs [37, p. 18] and [43, p. 29] it follows that such space is finite-dimensional. Lemma 3.2 is proved. Denote the linear space of centro-affine comitants of system (1.1)–(1.2) of type (3.1) by (d)

VΓ ,

(3.9)

and its dimension (d)

dimR VΓ .

(3.10)

48

The Center and Focus Problem

In the monograph [43, pp. 24–26], it is shown that for the spaces of comitants (3.9), there is a famous classic result (see., e.g., [16]). Theorem 3.1. The dimension of space (3.9) is determined by the equality (d)

dimR VΓ

= Ng − Ng−1 ,

(3.11)

where Ng (Ng−1 ) is equal to the set of all different systems of nonnegative integers 1

1

1

1

1

i













η 0 , η 1 , ..., η m1 , ξ 0 , ξ 1 , ..., ξ m1 , ..., η 0 , η 1 , ..., η m , ξ 0 , ξ 1 , ..., ξ m , among which there are also allowed the identical ones, satisfying system (3.6) together with the equation   

 i i i i i i 0η 0 + 1η 1 + ... + mi η mi + (−1)ξ 0 + 0ξ 1 + ... + (mi − 1)ξ mi = g,

i=0



     i i i i i i η 0 + 1η 1 + ... + mi η mi + (−1)ξ 0 + 0ξ 1 + ... + (mi − 1)ξ mi = g − 1 , i=0

(3.12)

where g is a weight of a comitant from space (3.9). Summing equations (3.6) with the first of (3.12), we obtain an equivalent system i

i

i

i

i

i

η 0 + η 1 + ... + η mi + ξ 0 + ξ 1 + ... + ξ mi = di , (i = 0, ), 

    i i i i i i η 0 + 2η 1 + ... + (mi + 1)η mi + ξ 1 + 2ξ 2 + ... + mi ξ mi = g + d , i=0

(3.13) where d is taken from  Definition 3.1. Consider the sum ug+d z0d0 z1d1 ...zd , which is written with the help of (3.13) as    i=0 ηi 0 +2ηi 1 +...+(mi +1)ηi mi +ξi 1 +2ξi 2 +...+mi ξi mi u ×

 

i

i

i

i

i

i

η 0 +η 1 +...+η mi +ξ 0 +ξ 1 +...+ξ mi

zi

.

i=0

Presenting this sum as    i=0

×

i

(uzi )η0



i

ziξ0





i

(u2 zi )η1 ... i

(uzi )ξ1 ...





i

(umi +1 zi )ηmi i

(umi zi )ξmi ,

Generating Functions and Hilbert Series

49

we obtain that it is equivalent to the product   i=0

mi +1 p=1

1 (1 − up zi )

1 . (1 − u q zi ) q=0

· mi

(3.14)

For more convenient use, the expression (3.14) can be written as  

(3.15)

Ψmi (u),

i=0

where

 Ψmi (u) =

1 (1−zi )(1−uzi )

for mi = 0,

1  mi (1−zi )(1−umi +1 zi ) k=1 (1−uk zi )2

for mi = 0

(3.16)

for each i = 0, 1, 2, ..., . From the above said, we obtain that the number Ng is equal to coefficient at ug+d z0d0 z1d1 z2d2 ...zd in decomposition of function (3.15) using (3.16) by degrees u, z0 , z1 , z2 , ..., z . Similarly, taking into account Theorem 3.1, it can be shown that the number Ng−1 is equal to coefficient at ug+d−1 z0d0 z1d1 ...zd in the decomposition by degrees u, z0 , z1 , ..., z of expression (3.15), taking into account (3.16), or, which is the same, to coefficient at ug+d z0d0 z1d1 ...zd in the expression  u i=0 Ψmi (u). From here, we have that difference Ng − Ng−1 at various g and d is a coefficient of the function ΨΓ (u) = (1 − u)

 

Ψmi (u),

(3.17)

i=0

which we call generating function for centro-affine comitants of system (1.1)– (1.2) with Γ = {mi }i=0 , where Ψmi (u) has the form (3.16). In this way, we conclude that the following is true: (d)

Theorem 3.2. dimR VΓ is equal to coefficient at the monomial ug+d z0d0 z1d1 ...zd in the decomposition of generating function (3.17) for centro-affine comitants of system (1.1)–(1.2) by positive degrees u, z0 , z1 , ..., z . Following A. Cayley (see, e.g., [1]), write expression (3.17), by replacing u by u−2 and zi by umi +1 zi in it, in the form (0) (0) (0) ϕΓ (u) = (1 − u−2 )ψm (u)ψm (u)...ψm (u), 0 1  (0)

where, according to (3.16), we have  1 (0) (u) = ψm i

(1−uzi )(1−u−1 zi )

1  mi (1−umi +1 zi )(1−u−mi −1 zi ) k=1 (1−umi −2k+1 zi )2

(3.18)

for mi = 0, for mi = 0 (3.19)

50

The Center and Focus Problem

for each Γ = {mi }i=0 . Following Sylvester (see, e.g., [1]), we will call expression (3.18) as initial form of the generating function for centro-affine comitants of system (1.1)– (1.2). Note that with the indicated substitutions A. Cayley, the monomial ug+d z0d0 z1d1 ...zd , taking into account equality (3.3), goes to the monomial uδ z0d0 z1d1 ...zd . Taking into account the last affirmation, as well as Theorem 3.2 and equalities (3.18)–(3.19), it is proved (d)

Theorem 3.3. dimR VΓ is equal to coefficient at the monomial d δ d0 d1 u z0 z1 ...z in the decomposition of initial form of generating function (3.18)–(3.19) for centro-affine comitants of system (1.1)–(1.2) by positive degrees u, z0 , z1 , ..., z . Example 3.3. If in system (1.1)–(1.2) we take Γ = {0, 1}, then we obtain  = 1, m0 = 0, m1 = 1. Then, from (3.18)–(3.19) for centro-affine comitants of system (1.18), we have an initial form of the generating function of the form (0)

ϕ0,1 (u) =

3.3

1 − u−2 . (1 − uz0 )(1 − u−1 z0 )(1 − u2 z1 )(1 − z1 )2 (1 − u−2 z1 )

(3.20)

Examples of Reduced Forms of Generating Functions for Centro-Affine Comitants of Differential Systems

In the papers of A. Cayley (see, for example, [1]) for binary forms, it is shown that if function (3.18)–(3.19) is represented as ϕΓ (u) − u−2 ϕΓ (u−1 ) = ϕΓ (u), (0)

(3.21)

then we can limit ourselves to studying only rational function ϕΓ (u), because the second term on the left side of (3.21) contains negative degrees of u, and according to Theorem 3.3 we are only interested in terms with positive degrees (0) in decomposition of the function ϕΓ (u). Note that from Theorem 3.3 and equality (3.21), it follows Proposition 3.1. ϕΓ (u) =

 (d)

(d)

dimR VΓ uδ z0d0 z1d1 ...zd .

(3.22)

Generating Functions and Hilbert Series

51

Example 3.4. For function (3.20) we find ϕ0,1 (u) − u−2 ϕ0,1 (u−1 ) = ϕ0,1 (u), (0)

(3.23)

where ϕ0,1 (u) =

1 + uz0 z1 . (1 − uz0 )(1 − z1 )(1 − z12 )(1 − z02 z1 )(1 − u2 z1 )

(3.24)

However, the question arises, how to obtain the function ϕΓ (u) from (3.21) for more complicated Γ. This problem is solved by applying the improved (0) Sylvester’s method [1] in decomposition of the function ϕΓ (u) in elementary fractions [14,33]. The resulting function ϕΓ (u) at Sylvester’s suggestion is named as reduced form of generating function. We illustrate this method with one example. Example 3.5. Suppose that in system (1.1)–(1.2), we have Γ = {2}, i.e.  = 0, m0 = 2. If in this case to denote z0 = c in (3.19), then from (3.18) for comitants of this system, we obtain a reduced form of generating function in the form (0)

ϕ2 (u) =

(1 −

u3 c)(1

1 + u−2 . − uc)2 (1 − u−1 c)2 (1 − u−3 c)

(3.25)

If you write the right part (3.25) in the form of elementary fractions with respect to u, then we obtain (0)

(1)

ϕ2 (u) = ϕ2 (u) + ϕ2 (u),

(3.26)

(1)

where ϕ2 (u) (ϕ2 (u)) is the sum of fractions corresponding to factors in the denominator of (3.25) with positive (negative) degrees of u. Then, according to this, we have ϕ2 (u) = A + B, (3.27) A=

2  i=0

 Ai Bj , B= , (1 − uc)2−j 1 − αi u j=0 1

(3.28)

and αi = i c1/3 (i = 0, 1, 2) and i are roots of the equation 3 − 1 = 0.

(3.29)

Multiplying both sides of equality (3.26) by 1 − u3 c and taking into account (3.27) and (3.28), we obtain ϕ2 (u)(1 − u3 c) =

2  i=0

Ai (1) (1 − u3 c) + [B + ϕ2 (u)](1 − u3 c). 1 − αi u

(3.30)

52

The Center and Focus Problem

Since 1 − u3 c = (1 − α0 u)(1 − α1 u)(1 − α2 u), then (3.30) can be written as (0)

ϕ2 (u)(1 − u3 c) =

2  i=0

Ai

2 

(1)

(1 − αj u) + [B + ϕ2 (u)](1 − u3 c).

(3.31)

j=0 j=i 

Substituting u = αi−1 into this equality, we obtain Ai

2 

(1 − αj αi−1 ) = ϕ2 (u)(1 − u3 c)|u=α−1 (i = 0, 1, 2). (0)

i

j=0  j=i

(3.32)

Taking into account that i are the roots of equation (3.29), it is easy to show that 2  (1 − αj αi−1 ) = 3 (i = 0, 1, 2). j=0 j=i 

Considering the last equality, from (3.32) and (3.25), we find 3Ai =

(1 −

1 − αi2 (i = 0, 1, 2). − αi c)(1 − αi3 c)

αi−1 c)2 (1

From here, taking into account that αi = i c1/3 (i = 0, 1, 2) and 3i = 1, we have 1 3Ai = (i = 0, 1, 2). (1 − c2 )(1 − αi2 )(1 − αi4 )2 Since 1 − αi2 =

1 − αi6 1 − αi12 , 1 − αi4 = , 2 4 1 + α i + αi 1 + αi4 + αi8

then from the last equality, we obtain Ai =

(1 + αi2 + αi4 )(1 + αi4 + αi8 )2 . 3(1 − c2 )2 (1 − c4 )2

Substituting (3.33) into the first equality from (3.28), we find

2   Ni 1 A= , 3(1 − c2 )2 (1 − c4 )2 i=0 1 − αi u

(3.33)

(3.34)

where Ni = (1 + αi2 + αi4 )(1 + αi4 + αi8 )2 (i = 0, 1, 2).

(3.35)

Bringing the expression in square brackets in (3.34) to a common denominator and taking into account that α0 + α1 + α2 = c1/3 (0 + 1 + 2 ) = 0,

Generating Functions and Hilbert Series

53

we will obtain 2 2 2 i=0 αi Ni + u (α1 α2 N0 + α0 α2 N1 + α0 α1 N2 ) i=0 Ni + u A= . 3(1 − c2 )2 (1 − c4 )2 (1 − u3 c)

(3.36)

Taking into account that 2 

 αim

=

i=0

3cn 0

for m = 3n, for m = 3n,

using (3.35), we find 2 

Ni = 3 + 6c2 + 15c4 + 3c6 ,

i=0 2 

αi Ni = 3c + 15c3 + 6c5 + 3c7 ,

i=0

α1 α2 N0 + α0 α2 N1 + α0 α1 N2 = 9c2 + 9c4 + 9c6 . From the last equalities and (3.36) after reduction by 3, we obtain A=

 1 1 + 2c2 + 5c4 + c6 + u(c + 5c3 (1 − c2 )2 (1 − c4 )2 (1 − u3 c) + 2c5 + c7 ) + u2 (3c2 + 3c4 + 3c6 ) .

(3.37)

If you multiply both parts (3.26) by (1−uc)2 , then using (3.25) and the second equality from (3.28), we will obtain 1 − u−2 . (1 − u3 c)(1 − u−1 c)2 (1 − u−3 c) (3.38) into this equality, we find (1)

B0 + B1 (1 − uc) + [A + ϕ2 (u)](1 − uc)2 = Substituting u = c−1

B0 =

c2 (1 −

c2 )2 (1

− c4 )

.

(3.39)

Taking the derivative with respect to u from both parts of (3.38) and taking u = c−1 , we have   d 1 − u− 2 −cB1 = , du (1 − u3 c)(1 − u−1 c)2 (1 − u−3 c) u=c−1 from where we obtain B1 = −

3c2 (1 + c2 + c4 ) . (1 − c2 )2 (1 − c4 )2

(3.40)

54

The Center and Focus Problem

Substituting (3.39) in (3.40), in the second equality of (3.28), we have B=

! " 1 −4c2 −3c4 −2c6 +u(3c3 +3c5 +3c7 ) . (3.41) (1 − c2 )2 (1 − c4 )2 (1 − uc)2

From (3.27) with the help of (3.40) and (3.41) after reducing by 1 − c2 , we obtain the following form of the generating function: ϕ2 (u) =

N2 (u, c) , D2 (u, c)

(3.42)

where D2 (u, c) = (1 − c2 )(1 − c4 )2 (1 − uc)2 (1 − u3 c), N2 (u, c) = 1 − c2 + c4 + u(−c + 3c3 − 2c5 ) + u2 (2c2 − 3c4 + c6 ) + u3 (−c3 + c5 − c7 ).

(3.43)

Example 3.6. Suppose that in system (1.1)–(1.2), we have Γ = {3}, i.e.  = 0, m0 = 3. If in this case to denote z1 = d in (3.19), then from (3.18) for comitants of this system, we obtain the reduced form of generating function of the form (0)

ϕ3 (u) =

(1 −

u4 d)(1

−

1 − u− 2 . − d)2 (1 − u−2 d)2 (1 − u−4 d)

u2 d)2 (1

(3.44)

Analogously to Example 3.5, using this function, we obtain the following reduced form of generating function: ϕ3 (u) =

N3 (u, d) , D3 (u, d)

(3.45)

where D3 (u, d) = (1 − d2 )3 (1 − d3 )2 (1 − u2 d)2 (1 − u4 d), N3 (u, d) = 1 − d2 + d4 + u2 (−d + d2 + 3d3 − 2d5 ) + u4 (2d2 − 3d4 − d5 + d6 ) + u6 (−d3 + d5 − d7 ).

(3.46)

Observation 3.3. In the papers [17], [18], [33], [34], the reduced forms of generating functions for comitants of differential systems (1.1)–(1.2) for Γ = {0}, {1}, {0, 1}, {2}, {0, 2}, {1, 2}, {0, 1, 2}, {3}, {0, 3}, {1, 3}, {2, 3}, {4}, {1, 4}, {5}, {1, 5} are found. Observation 3.4. The reduced form of generating function for invariants of differential system (1.1)–(1.2) with fixed Γ is obtained from the reduced form of generating function for comitants ϕΓ (u) of the same system for u = 0.

Generating Functions and Hilbert Series

3.4

55

Hilbert Series for Graded Algebras of Unimodular Comitants and Invariants of Differential Systems

Let SL(2, R) ⊆ GL(2, R) be a subgroup of unimodular transformations, i.e. SL(2, R) consists of transformations of the form (1.27), for which Δ = 1. Definition 3.2. We say that an integer rational function 0

0

0







2 L(x, y, a10 , a11 , ..., a1m0 , ..., a02 , a12 , ..., am ) 

with respect to the phase variables x, y and coefficients of system (1.1)–(1.2) is called a unimodular comitant of this system if the following equality takes place: 0

0

0







0

0

0







1 L(x, y, b10 , b11 ,...,bm ,...,b20 , b21 ,...,b2m ) = L(x, y, a10 , a11 ,...,a1m0 ,...,a20 , a21 ,...,a2m ) 0

for any coefficients of system (1.1)–(1.2), phase variables x, y and transformations q ∈ SL(2, R) from (1.27). Similarly to (3.9) we denote the linear space of unimodular comitants of system (1.1)–(1.2) of type (3.1) by (d)

(3.47)

SΓ . (d) (d) Lemma 3.3. VΓ ∼ = SΓ . Proof. Let comitant 0

0

0







2 ) K(x, y, a10 , a11 , ..., a1m0 , ..., a20 , a12 , ..., am 

belong to space (3.9). Then, taking into account the inclusion SL(2, R) ⊆ GL(2, R), the mentioned comitant will be an element of spaces (3.47), which indicates a one-to-one correspondence in one direction between the elements of spaces (3.9) and (3.47). Let us show that such correspondence exists in the contrary direction. To do this, use Remark 1.2. If in system (1.1)–(1.2) one realizes transformation (1.28), then for coefficients of system (2.19) we will have i

i

1

bkj = Δ− 2 (mi −1) akj . Then, according to the fact that K is a homogeneous comitant of type (3.1) for unimodular comitant L that belongs to space (3.47), with this transformation we will have equality 1

1

1

0

1

1

0

0

L(Δ 2 x, Δ 2 y, Δ− 2 (m0 −1) a10 , Δ− 2 (m0 −1) a11 , ..., Δ− 2 (m0 −1) a0m0 , 1

1

1

1

0

0

1



Δ− 2 (m0 −1) a20 , Δ− 2 (m0 −1) a21 , .., Δ− 2 (m0 −1) a2m0 , ..., Δ− 2 (m −1) a2m ) = 0

0

0







= Δ−g K(x, y, a10 , a11 , ..., a1m0 , ..., a20 , a21 , ..., a2m ),

(3.48)

56

The Center and Focus Problem

where for g we have equality (3.13). If in the resulting system after transformation (1.28) one realizes unimodular transformation (1.29), then for unimodular comitant from the left side of (3.48) we will have 0

0

0







2 L(x, y, b10 , b11 , ..., b1m0 , ..., b02 , b12 , ..., bm )=   1 1 1 1 1 0 0 0 = L Δ 2 x, Δ 2 y, Δ− 2 (m0 −1) a10 , Δ− 2 (m0 −1) a11 , ..., Δ− 2 (m0 −1) a1m0 ,

 1 1 1 1 0 0 0  Δ− 2 (m0 −1) a02 , Δ− 2 (m0 −1) a21 , .., Δ− 2 (m0 −1) a2m0 , ..., Δ− 2 (m −1) a2m . Then, using the last equality and taking into account (3.48), we have the identity 0

0

0







L(x, y, b10 , b11 , ..., b1m0 , ..., b02 , b12 , ..., b2m ) = 0

0

0







= Δ−g K(x, y, a10 , a11 , ..., a1m0 , ..., a20 , a21 , ..., a2m ) for any transformation (1.28), for any coefficients of system (1.1)–(1.2) and any phase variables x, y. Note that g is defined using (3.3) and, according to Remark 2.1, it is an integer. Thus, it was established that a unimodular comitant of type (3.1) is uniquely associated with a centro-affine comitant of system (1.1)–(1.2), which coincides with it. Lemma 3.3 is proved. The proof of Lemma 3.3 contains a criterion for the centro-affine invariance of any homogeneous polynomial in the coefficients of system (1.1)–(1.2) and the phase variables x, y, which is formulated as Consequence 3.2. In order that an integer rational and homogeneous function of type (3.1) in coefficients of system (1.1)–(1.2) to be a centro-affine comitant of this system, it is necessary and sufficient that it be a unimodular comitant of the same type (3.1) of the mentioned system. Consequence 3.3. The following equality takes place: (d)

dimR VΓ Consider a linear space

SΓ =

(d)

= dimR SΓ . 

(d)

SΓ ,

(3.49)

(d)

which is a graded algebra of comitants, in which its components satisfy the (d) (e) (d+e) . inclusion SΓ SΓ ⊆ SΓ Following the paper [41], under a generalized Hilbert series of algebra (3.49), we will understand  (d) H(SΓ , u, z0 , z1 , ..., z ) = dimR SΓ uδ z0d0 z1d1 ...zd . (3.50) (d)

Generating Functions and Hilbert Series

57

From equalities (3.22) and (3.50) with the help of Consequence 3.3 we obtain H(SΓ , u, z0 , z1 , ..., z ) = ϕΓ (u).

(3.51)

Note that (according to the same paper [41]) an ordinary Hilbert series is obtained in an obvious way from the generalized HSΓ (u) = H(SΓ , u, u, u, ..., u).

(3.52)

Thus, using (3.51), we have Conclusion 3.1. The reduced form of generating function for comitants of system (1.1)–(1.2) with a fixed Γ is a generalized Hilbert series for algebra of unimodular comitants (3.49) with the same Γ. Observation 3.5. If we denote the algebra of invariants for a fixed Γ for system (1.1)–(1.2) by SIΓ , then according to Observation 3.4 and equality (3.51) for generalized Hilbert series of this algebra, we have H(SIΓ , z0 , z1 , ..., z ) = H(SΓ , 0, z0 , z1 , ..., z ) = ϕΓ (0),

(3.53)

and for the ordinary Hilbert series, we obtain HSIΓ (z) = H(SIΓ , z, z, ..., z).

(3.54)

Important Remark 3.1. Note that centro-affine comitants and invariants of the systems of the form (1.1)–(1.2) were first studied in the works of Academician K. S. Sibirsky [37–39] and further developed in the works of his students. Since in the present section it is shown that the mentioned comitants and invariants form a basis of the graded algebras of comitants SΓ and invariants SIΓ , then these algebras will be called Sibirsky graded algebras of comitants SΓ and Sibirsky graded algebras of invariants SIΓ for the system s(Γ). In the future we will use the abbreviation ”Sibirsky algebras” SΓ and SIΓ .

3.5

Comments to Chapter Three

It is worth noting that the method for generating functions is hundreds of years old. It was used in the papers of I. Newton (1642– 1727), D. Bernoulli (1700–1782), L. Euler (1707–1783), K. Gauss (1777–1855), R. Riemann (1826–1866), A. Cayley (1821–1895), D. Sylvester (1814–1897), D. Hilbert (1862–1943) and other scientists to prove unexpected results. Probably, the first manifestation of this method is the Newton binomial formula, which says that the number  n n! = k k!(n − k)!

58

The Center and Focus Problem

is the coefficient of tk in the polynomial (1 + t)n , i.e. n   n k n (1 + t) = t . k k=0

In modern language, we can say that the function (1 + t)n is a generating function for numbers    n n n , , ..., . 0 1 n From these considerations, such numbers are also called binomial coefficients. The method of generating functions is based on a very simple idea. A sequence of real numbers a0 , a1 , a2 , ... is associated with an expression of the form a(t) = a0 + a1 t + a2 t2 + ..., which we will call a series, or a generating function of this sequence. This function can be represented as a polynomial of infinite degree. Such an expression is called a formal power series, since we are not interested in its convergence. These series often have simple forms that allow us to draw certain conclusions about the sequence {an }n≥0 , which are very difficult to obtain in another way. Let V be a vector space represented as a direct sum of finite-dimensional subspaces ∞ $ # V = V n , Vn Vm = {0}. (n=m) 

n=0

Such expansion will be called a graduation. A generating function for V , or sequence dimR Vn (n = 0, 1, 2, 3, ...), will be called the formal series ΦV (t) =

∞ 

(dimR Vn )tn .

(3.55)

n=0

A remarkable effect for generating functions is that the corresponding formal series can converge in some neighborhood of zero to a concrete function. Thus, by studying its properties (e.g., poles, zeros), we can obtain additional information on the structure of the space V , in particular, on the asymptotic behavior of the sequence {dimR Vn }∞ n=0 . If V = A is a graded algebra, then (3.55) is called the Hilbert series for this algebra and is denoted by HA (t), which carries profound information on the nature of asymptotic behavior of the algebra A. By studying the space V or the algebra A, the generating functions or the Hilbert series, which depend on several variables, can be introduced in some cases. This fact reflects a more detailed graduation of these objects. As a result, these functions are called the generalized generating functions and the Hilbert series, respectively, and those having the form (3.55), are called ordinary.

Generating Functions and Hilbert Series

59

The application of generating functions and Hilbert series in the theory of two-dimensional autonomous first-order polynomial differential systems has its origin in the papers [33,34]. We note that (see [41]) the term Hilbert series arises from the classical Hilbert results relating to the commutative case. Sometimes it is also called the Poincar´e series, but today this term should be considered to be settled, linking the name of Poincar´e only to the homologous series. Despite the fact that the algebras SΓ and SIΓ for system of the form (1.1)–(1.2) have their origin and are thoroughly studied in the papers [33,34], they received the name of ”Sibirsky algebra” only in the article [29]. This was because one of the most important problems in the qualitative theory of differential systems, the problem of the number of algebraically independent focus quantities, could be solved using these algebras, which are involved in solving the center and focus problem for any differential system of the form (1.1)–(1.2) with polynomial nonlinearities.

4 Hilbert Series for Sibirsky Algebras SΓ (SIΓ) and Krull Dimension for Them

4.1

Krull Dimension for Sibirsky Graded Algebras

In the following, we will limit ourselves to studying systems of the form s(1, m1 , m2 , ..., m` ) from (1.1)–(1.2) and therefore the Sibirsky algebras S1,m1 ,m2 ,...,m` and SI1,m1 ,m2 ,...,m` . From theory of invariants of differential systems [37] and tensors [16], it follows that Sibirsky graded algebras S1,m1 ,m2 ,...,m` and SI1,m1 ,m2 ,...,m` are commutative and finitely defined algebras. If for these algebras, we introduce a single notation A, then the last statement for them is written in the form A =< a1 , a2 , ..., am |f1 = 0, f2 = 0, ..., fn = 0 > (m, n < ∞),

(4.1)

where ai are the generators of this algebra, fj – defining relations (syzygies) between these generators. It is known, for example, from [33], that for the simplest differential system s(0, 1) from (1.3) of the form x˙ = a + cx + dy, y˙ = b + ex + f y the finitely defined graded algebras of comitants S0,1 and invariants SI0,1 will be written, respectively, as S0,1 =< i1 , i2 , i3 , k1 , k2 , k3 |(i1 k1 − k3 )2 + k32 − i2 k12 − 2i3 k2 = 0 >, SI0,1 =< i1 , i2 , i3 >,

(4.2)

where according to (2.12), we have i1 = c + f, i2 = c2 + 2de + f 2 , i3 = −ea2 + (c − f )ab + db2 , k1 = −bx + ay, k2 = −ex2 + (c − f )xy + dy 2 , k3 = −(ea + f b)x + (ca + db)y. Note that in [28] on example of the system s(0, 1) from (1.3), some experience was gained in the application of classical groups, Lie algebras and theory DOI: 10.1201/9781003193074-5

61

62

The Center and Focus Problem

of invariants and comitants in the qualitative study of autonomous polynomial differential systems, accumulated in the Chi¸sin˘au school of differential equations. The work [33] also contains examples of Sibirsky algebras of invariants for systems (1.7) and (1.10), which are written, respectively, as SI2 =< I7 , I8 , I9 , I15 |f1 = 0 >, where I7 − I9 , I15 are from (2.65) and f1 is from (2.67), and SI3 =< J1 , J2 , J3 , J4 , J5 , J6 |ϕ1 = 0 >, where J1 − J6 are from (2.71) and ϕ1 is from (2.73). Definition 4.1. [3] The elements a1 , a2 , ..., ar of the algebra A are called algebraically independent, if for any nontrivial polynomial F in r variables, the following inequality holds:  0. F (a1 , a2 , ..., ar ) = Definition 4.2. Maximal number of algebraically independent elements of a graded algebra A is called the Krull dimension of this algebra, which is denoted by (A). It is known that for the algebra A, given in the form (4.1), the equality n = m − (A) is true. However, this equality is not very effective, since it is impossible to determine the numbers m and n for the majority of algebras of invariants and comitants of systems of the form (1.1)–(1.2). In the classical theory of invariants [1] the set of elements a1 , a2 , ..., a (A) from A, which reflect the Krull dimension of the algebra A, are called algebraic  basis of this set. This means (similar to Remark 2.2) that for ∀a ∈ A (a = aj ) there is such a natural number p, corresponding to a, that the following equality holds: (4.3) P0 ap + P1 ap−1 + ... + Pp = 0, where Pk (k = 0, p are polynomials on aj (j = 1, (A)). Note that generally speaking P0 ≡ 1. If for all a ∈ A in (4.3), we have P0 ≡ 1, then such basis is called integral algebraic. Its existence is shown by Hilbert (see, e.g. [1]). The number of its elements is denoted by  (A). Note that the number of elements in the mentioned bases does not always coincide between themselves. So in [33], we have that for the system s(4) the Krull dimension (SI4 ) = 7, and in [42] for the same system, we find that the number of elements in integral algebraic basis of the same algebra is  (SI4 ) = 9, i.e. (SI4 ) <  (SI4 ). From [33], we have that for the system s(0, 1) the equalities (S0,1 ) =  (S0,1 ) = 5 and (SI0,1 ) = =  (SI0,1 ) = 3 are true. From the papers [14,33,44] it follows that for the systems s(2) and s(3)

Hilbert Series for Sibirsky Algebras

63

we have (SI2 ) =  (SI2 ) = 3, (SI3 ) = =  (SI3 ) = 5. In the papers [21,33] we find that for the system s(1, 2) the equality (SI1,2 ) =  (SI1,2 ) = 7 is true. However, for the system s(1, 2, 3), according to [33] and [4], we have that (SI1,2,3 ) = 15 and  (SI1,2,3 ) = 21. These examples lead us to (A) ≤  (A). This inequality underlines that an integral algebraic basis contains an algebraic basis of the algebra A. The proof of this fact is easily obtained by contradiction. Remark 4.1. The main property of integral algebraic basis of the algebra of invariants A is that this is the smallest number of elements of the algebra A, the equity to zero of which turns to zero all elements of the algebra A. Further, we will need obvious statements Proposition 4.1. If B is a graded subalgebra of the algebra A, then between the Krull dimensions of these algebras, the inequality holds (B) ≤ (A) . Proposition 4.2. If the Krull dimension of the algebra A is (A), then on any variety V = {a = 0, b < 0} with fixed a, b ∈ A (b does not affect the mentioned variety) in the algebra A the number of algebraically independent elements of this algebra is not greater then (A) (maybe the number of elements that form an integral algebraic basis is not greater than (A)). From Theorem 2.4, Remark 2.2, and Definition 4.2, it follows Conclusion 4.1. Krull dimension for Sibirsky graded algebras S1,m1 ,m2 ,...,m and SI1,m1 ,m2 ,...,m are expressed by the formulas

   (4.4) mi +  + 3, (S1,m1 ,m2 ,...,m ) = 2 i=1

(SI1,m1 ,m2 ,...,m ) = 2

 

 mi + 

+ 1.

(4.5)

i=1

4.2

Hilbert Series for Sibirsky Graded Algebras S1,m1 ,m2 ,...,m , SI1,m1 ,m2 ,...,m

According to Theorem 3.3 and Consequence 3.3, we obtain that in Sibirsky (d) algebra S1,m1 ,m2 ,...,m for its spaces, we have dimR S1,m1 ,m2 ,...,m < ∞.

64

The Center and Focus Problem

Then, the following (3.50), by the generalized Hilbert series of the algebra S1,m1 ,m2 ,...,m , we will understand the formal series H(S1,m1 ,m2 ,...,m , u, z0 , z1 , ..., z )  (d) = dimR S1,m1 ,m2 ,...,m uδ z0d0 z1d1 ...zd ,

(4.6)

(d)

about which it is said that it reflects u, z – graduation of the mentioned algebra. According to Observation 3.5 for the algebra of invariants SI1,m1 ,m2 ,...,m , we obtain the equality H(SI1,m1 ,m2 ,...,m , z0 , z1 , ..., z ) = H(S1,m1 ,m2 ,...,m , 0, z0 , z1 , ..., z ),

(4.7)

and usual Hilbert series are written, respectively, as HS1,m1 ,m2 ,...,m (u) = H(S1,m1 ,m2 ,...,m , u, u, u, ..., u), HSI1,m1 ,m2 ,...,m (z) = H(SI1,m1 ,m2 ,...,m , z, z, ..., z).

(4.8)

The last series carry important information about the nature of asymptotic behavior of these algebras. The method of constructing generalized Hilbert series (4.6)–(4.8) for Sibirsky algebras S1,m1 ,m2 ,...,m and SI1,m1 ,m2 ,...,m is developed in the papers [33,34] and shown in simple examples from Section 3.3. As it was shown in this section, the generalized Hilbert series for algebras S0,1 and SI0,1 of unimodular comitants and invariants of the system s(0, 1) have the form, respectively, 1 + uz0 z1 , (1 − uz0 )(1 − z1 )(1 − z12 )(1 − z02 z1 )(1 − u2 z1 ) 1 . H(SI0,1 , z0 , z1 ) = (1 − z1 )(1 − z12 )(1 − z02 z1 )

H(S0,1 , u, z0 , z1 ) =

Then, according to (4.8), with their help, an ordinary Hilbert series is written as HS0,1 (u) =

(1 −

1 − u + u2 1 , HSI0,1 (z) = . − u2 )(1 − u3 )2 (1 − z)(1 − z 2 )(1 − z 3 )

u)2 (1

Remark 4.2. Following [40], note that the Krull dimension (S1,m1 ,m2 ,...,m ) ((SI1,m1 ,m2 ,...,m )) for Sibirsky graded algebras S1,m1 ,m2 ,...,m (SI1,m1 ,m2 ,...,m ) equals to order of pole of the ordinary Hilbert series HS1,m1 ,m2 ,...,m (u) (HSI1,m1 ,m2 ,...,m (z)) at the unit. For example, taking into account the above-mentioned Hilbert series HS0,1 (u) and HSI0,1 (z), we obtain for Krull dimension of the algebras S0,1 and SI0,1 , respectively, (S0,1 ) = 5 and (SI0,1 ) = 3.

Hilbert Series for Sibirsky Algebras

65

In the other cases, when there is no explicit form of ordinary Hilbert series, but a power series decomposition is known, the following can be used: Remark 4.3. We agree that the  comparison  ofn series with non-negative coefficients occurs by coefficients ( an tn ≤ bn t ⇔ an ≤ bn , ∀n). Taking this into account, if for commutative graded algebras A and B, we have HA (t) ≤ HB (t),

(4.9)

then for their Krull’s dimensions we also obtain (A) ≤ (B). It is also clear that if for ordinary Hilbert series of the commutative graded algebra A, we have C HA (t) ≤ , (4.10) (1 − t)m where C – some fixed constant, then we obtain (A) ≤ m. The proof of Remark 4.3 is obtained using the Macaulay theorem from the paper [13]. Consequence 4.1. Note that formulas (4.4)–(4.5) contain an explicit form of Krull dimensions for Sibirsky graded algebras S1,m1 ,m2 ,...,m , SI1,m1 ,m2 ,...,m of the differential system s(1, m1 , m2 , ..., m ). However, knowledge of Hilbert series for these algebras gives additional information about the mentioned algebras, which will be used further. At the same time, using these series once again, an information about Krull dimensions of Sibirsky algebras for concrete systems s(1, m1 , m2 , ..., m ) is confirmed.

4.3

Hilbert Series for Sibirsky Algebras S1,2 , SI1,2 and Their Krull Dimensions

From (3.18)–(3.19) for Γ = {1, 2}, putting  = 1 and m0 = 1, m1 = 2 and introducing for convenience the notation z0 = b, z1 = c, we find for comitants of the differential system with quadratic nonlinearities s(1, 2) an initial form of generating function ϕ1,2 (u) = (1 − u−2 )ψ1 (u)ψ2 (u),

(4.11)

1 , − b)2 (1 − u−2 b) (1 − 1 (0) . ψ2 (u) = (1 − u3 c)(1 − uc)2 (1 − u−1 c)2 (1 − u−3 c)

(4.12)

(0)

where

(0)

(0)

(0)

ψ1 (u) =

u2 b)(1

66

The Center and Focus Problem

Using the advanced Sylvester method for decomposition of the function (0) ϕ1,2 (u) on elementary fractions by analogy with the Example 3.5 and taking into account Cayley functional equation (3.21) and Conclusion 3.1, we obtain Theorem 4.1. A generalized Hilbert series H(S1,2 , u, b, c) for Sibirsky algebra S1,2 of the system s(1, 2) from (1.5) is a rational function of u, b, c and has the form N1,2 (u, b, c) H(S1,2 , u, b, c) = , (4.13) D1,2 (u, b, c) where D1,2 (u, b, c) = (1 − b)(1 − b2 )(1 − c2 )(1 − c4 )2 (1 − bc2 )2 (1 − b3 c2 )· ·(1 − u2 b)(1 − uc)2 (1 − u3 c),

(4.14)

N1,2 (u, b, c) = 1 − c2 + c4 + b(c2 + 2c4 − 2c6 ) + b2 (c2 + c4 − c6 − c8 ) +b3 (2c4 − 2c6 − c8 ) + b4 (−c6 + c8 − c10 ) + u[−c + 3c3 − 2c5 +b(2c − 5c5 + 3c7 ) + b2 (c − 2c5 + c9 ) + b3 (c3 − 3c5 + 2c7 ) + b4 (c7 −3c9 + 2c11 )] + u2 [2c2 − 3c4 + c6 + b(−3c4 + 4c6 − 2c8 ) + b2 (−2c4 +c10 ) + b3 (c2 − 3c4 + 2c10 ) + b4 (−2c4 + 2c6 − c8 + 3c10 − c12 ) +b5 (c6 − c8 + c10 )] + u3 [−c3 + c5 − c7 + b(c − 3c3 + c5 − 2c7 + 2c9 ) +b2 (−2c3 + 3c9 − c11 ) + b3 (−c3 + 2c9 ) + b4 (2c5 − 4c7 + 3c9 ) +b5 (−c7 + 3c9 − 2c11 )] + u4 [b(−2c2 + 3c4 − c6 ) + b2 (−2c6 + 3c8 − c10 )+ +b3 (−c4 + 2c8 − c12 ) + b4 (−3c6 + 5c8 − 2c12 ) + b5 (2c8 − 3c10 + c12 )] +u5 [b(c3 − c5 + c7 ) + b2 (c5 + 2c7 − 2c9 ) + b3 (c5 + c7 − c9 − c11 ) +b4 (2c7 − 2c9 − c11 ) + b5 (−c9 + c11 − c13 )]. (4.15) Proof of Theorem 4.1 follows from the validity of the Cayley functional equation H(S1,2 , u, b, c) − u−2 H(S1,2 , u−1 , b, c) = ϕ1,2 (u), (0)

(0)

where H(S1,2 , u, b, c) is from (4.13)–(4.15), and ϕ1,2 (u) is from (4.11)–(4.12). According to Observation 3.5 of Theorem 4.1, we have Consequence 4.2. A generalized Hilbert series H(SI1,2 , b, c) for Sibirsky algebra SI1,2 has the form H(SI1,2 , b, c) =

N I1,2 (b, c) , DI1,2 (b, c)

(4.16)

Hilbert Series for Sibirsky Algebras

67

where DI1,2 (b, c) = (1 − b)(1 − b2 )(1 − c2 )(1 − c4 )2 (1 − bc2 )2 (1 − b3 c2 ), N I1,2 (b, c) = 1 − c2 + c4 + b(c2 + 2c4 − 2c6 ) + b2 (c2 + c4 − c6

(4.17)

−c ) + b (2c − 2c − c ) + b (−c + c − c ). 8

3

4

6

8

4

6

8

10

Using equalities (3.52) and (3.54) from expressions (4.13)–(4.17), we obtain that there takes place Theorem 4.2. Ordinary Hilbert series HS1,2 (u) and HSI1,2 (z) for Sibirsky algebras S1,2 and SI1,2 of the system s(1, 2) from (1.5) have the form HS1,2 (u) =

1 (1 −

u2 )2 (1

−

u3 )3 (1

− u4 )3 (1 − u5 )

(1 + u + u2

+ 4u3 + 11u4 + 20u5 + 29u6 + 33u7 + 39u8 + 41u9 + 39u10 + 33u11 + 29u12 + 20u13 + 11u14 + 4u15 + u16 + u17 + u18 ), (4.18) HSI1,2 (z) =

1 + z 3 + 2z 4 + 3z 5 + 3z 6 + 3z 7 + 2z 8 + z 9 + z 12 . (1 − z)(1 − z 2 )(1 − z 3 )2 (1 − z 4 )2 (1 − z 5 )

(4.19)

Theorem 4.3. Krull dimensions of Sibirsky algebras S1,2 and SI1,2 are equal to (S1,2 ) = 9 and (SI1,2 ) = 7, respectively. Proof. From the paper [22], it is known that in order that a point u = 1 (z = 1), to be a pole of the function HSΓ (u) HSIΓ (z) with order multiplicities k (k ≥ 1), it is necessary andsufficient that it be a zero of multiplicity k for  1 1 the function HS (u) HSI (z) . Γ

Γ

We illustrate this with the ordinary Hilbert series HS1,2 (u) and HSI1,2 (z) from (4.18) and (4.19). It is easy to see that (1 − u)9 (1 + + u + u2 )−3 (1 + u + u2 + u3 )−3 HS1,2 (u)  1 · 1 + u + u2 + 4u3 + 11u4 + 20u5 + 29u6 2 3 4 −1 (1 + u + u + u + u ) 1

=

u)−2 (1

+ 33u7 + 39u8 + 41u9 + 39u10 + 33u11 + 29u12 + 20u13 + 11u14 + 4u15 − 1 + u16 + u17 + u18 and (1 − z)7 (1 + z)−1 (1 + z + z 2 )−2 (1 + z + z 2 + z 3 )−2 HSI1,2 (z)  1 · 1 + z 3 + 2z 4 + 3z 5 + 3z 6 + 3z 7 + 2z 8 (1 + z + z 2 + z 3 + z 4 )−1 −1 , + z 9 + z 12 1

=

68

The Center and Focus Problem

from where we find that limu→1 (1 − u)9 HS1,2 (u) =  0 and limz→1 (1 − z)7 · ·HSI1,2 (z) = 0. Hence we have that at the point u = 1 (z = 1) the function HS1,2 (u) (HSI1,2 (z)) has a pole of multiplicity 9 (7). According to Remark 4.2, Theorem 4.3 is proved.

Hilbert Series for Sibirsky Algebras S1,3 , SI1,3 and Their Krull Dimensions

4.4

From (3.18)–(3.19) for Γ = {1, 3}, by accepting  = 1 and m0 = 1, m1 = 3 and introducing for convenience the notation z0 = b, z1 = d, we find for comitants of the differential system s(1, 3) the initial form of the generating function ϕ1,3 (u) = (1 − u−2 )ψ1 (u)ψ3 (u), (0)

(0)

(0)

(4.20)

where 1 , (1 − u2 b)(1 − b)2 (1 − u−2 b) 1 (0) ψ3 (u) = . (1 − u4 d)(1 − u2 d)2 (1 − d)2 (1 − u−2 d)2 (1 − u−4 d) (0)

ψ1 (u) =

(4.21)

(0)

Using Sylvester’s advanced method of decomposition of the function ϕ1,3 (u) to elementary fractions, by analogy with Example 3.5 and taking into account Cayley functional equation (3.21) and Conclusion 3.1, we obtain that the following statement is true Theorem 4.4. A generalized Hilbert series H(S1,3 , u, b, d) for Sibirsky algebra S1,3 of the system s(1, 3) from (1.8) is a rational function of u, b, d and has the form H(S1,3 , u, b, d) =

N1,3 (u, b, d) , D1,3 (u, b, d)

(4.22)

where D1,3 (u, b, d) = (1 − b)(1 − b2 )(1 − u2 b)(1 − bd)2 (1 − b2 d)(1 − d2 )3 ·(1 − d3 )2 (1 − u2 d)2 (1 − u4 d),

N1,3 (u, b, d) =

4  k=0

R2k (b, d)u2k ,

(4.23)

Hilbert Series for Sibirsky Algebras

69

and R0 (b, d) = 1 + b(−d + d2 + 3d3 − 2d5 ) + b2 (2d2 − 3d4 − d5 + d6 ) +b3 (−d3 + d5 − d7 ) − d2 + d4 , R2 (b, d) = b(2d + 4d2 − 2d3 − 8d4 + 4d6 ) + b2 (d − d2 − 6d3 + 7d5 − 3d7 ) +b3 (−2d2 + 4d4 − 4d6 + 2d8 ) + b4 (d3 − d5 + d7 ) − d + d2 + 3d3 − 2d5 , R4 (b, d) = b(d − d2 − 6d3 + 7d5 − 3d7 ) + b2 (−3d2 − 2d3 + 6d4 − 6d6 +2d7 + 3d8 ) + b3 (3d3 − 7d5 + 6d7 + d8 − d9 ) + b4 (−d4 + d5 + 3d6 − 2d8 ) +2d2 − 3d4 − d5 + d6 , R8−2k (b, d) = −b4 d10 R2k (b−1 , d−1 ) (k = 0, 1). (4.24) According to Observation 3.5 from Theorem 4.4, we have Consequence 4.3. A generalized Hilbert series H(SI1,3 , b, d) for Sibirsky algebra SI1,3 has the form H(SI1,3 , b, d) =

N I1,3 (b, d) , DI1,3 (b, d)

(4.25)

where DI1,3 (b, d) = (1 − b)(1 − b2 )(1 − bd)2 (1 − b2 d)(1 − d2 )3 (1 − d3 )2 , N I1,3 (b, d) = 1 − d2 + d4 + b(−d + d2 + 3d3 − 2d5 ) + b2 (2d2 − 3d4 − d5 + d6 ) + b3 (−d3 + d5 − d7 ).

(4.26)

Using equalities (3.52) and (3.54) from expressions (4.22)–(4.26), we obtain that there takes place the following Theorem 4.5. Ordinary Hilbert series HS1,3 (u) and HSI1,3 (z) for Sibirsky algebra S1,3 and SI1,3 of the system s(1, 3) from (1.8) have the form HS1,3 (u) =

 1 1 + u + u3 + 9u4 + 16u5 (1 − u2 )5 (1 − u3 )5 (1 − u5 )

+ 19u6 + 15u7 + 14u8 + 15u9 + 19u10 + 16u11 + 9u12 + u13  (4.27) + u15 + u16 ,  1 1 − z 2 + z 3 + 5z 4 + z 5 2 5 3 3 (1 − z)(1 − z ) (1 − z )  − z6 + z8 .

HSI1,3 (z) =

(4.28)

Theorem 4.6. Krull dimensions of Sibirsky algebras S1,3 and SI1,3 are equal to (S1,3 ) = 11 and (SI1,3 ) = 9, respectively . The proof of this theorem is similar to the proof of Theorem 4.3.

70

The Center and Focus Problem

4.5

Hilbert Series for Sibirsky Algebras S1,4 , SI1,4 and Their Krull Dimensions

From (3.18)–(3.19) for Γ = {1, 4}, by accepting  = 1 and m0 = 1, m1 = 4 and introducing for convenience the notation z0 = b, z1 = e, we find for comitants of the differential system s(1, 4) the initial form of the generating function ϕ1,4 (u) = (1 − u−2 )ψ1 (u)ψ4 (u), (0)

(0)

(0)

(4.29)

where (0)

ψ1 (u) = (0)

ψ4 (u) =

1 , (1 − u2 b)(1 − b)2 (1 − u−2 b) 1 . (1 − u5 e)(1 − u3 e)2 (1 − ue)2 (1 − u−1 e)2 (1 − u−3 e)2 (1 − u−5 e) (4.30) (0)

Using Sylvester’s advanced method of decomposition of the function ϕ1,4 (u) to elementary fractions, by analogy with Example 3.5 and taking into account Cayley functional equation (3.21) and Conclusion 3.1, we obtain that the following statement is true Theorem 4.7. Ordinary Hilbert series H(S1,4 , u, b, e) for Sibirsky algebra S1,4 of the system s(1, 4) is a rational function of u, b, e and has the form H(S1,4 , u, b, e) =

N1,4 (u, b, e) , D1,4 (u, b, e)

(4.31)

where D1,4 (u, b, e) = (1 − b)(1 − b2 )(1 − bu2 )(1 − be2 )2 (1 − b3 e2 )2 (1 − b5 e2 ) ·(1 − e2 )(1 − e4 )2 (1 − e6 )2 (1 − e8 )2 (1 − eu)2 (1 − eu3 )2 (1 − eu5 ), N1,4 (u, b, e) =

13 

(4.32)

Rk (b, e)uk ,

k=0

and Rk is from Appendix 1. According to Observation 3.5 from Theorem 4.7, we have Consequence 4.4. Ordinary Hilbert series H(SI1,4 , b, e) for Sibirsky algebra SI1,4 has the form N I1,4 (b, e) H(SI1,4 , b, e) = , (4.33) DI1,4 (b, e) where DI1,4 (b, e) = (1 − b)(1 − b2 )(1 − be2 )2 (1 − b3 e2 )2 (1 − b5 e2 )(1 − e2 )(1 − e4 )2 · (1 − e6 )2 (1 − e8 )2 , N I1,4 (b, e) = R0 (b, e),

(4.34)

Hilbert Series for Sibirsky Algebras

71

and R0 (b, e) is from Appendix 1. Using equalities (3.52) and (3.54) from expressions (4.31)–(4.34), we obtain that there takes place Theorem 4.8. Ordinary Hilbert series HS1,4 (u) and HSI1,4 (z) for Sibirsky algebras S1,4 and SI1,4 of system s(1, 4) have the form HS1,4 (u) =

N1,4 (u) , D1,4 (u)

(4.35)

where D1,4 (u) = (1 − u2 )(1 − u3 )(1 − u4 )3 (1 − u5 )2 (1 − u6 )3 (1 − u7 )(1 − u8 )2 , (4.36) N1,4 (u) = 1 + u + u2 + 5u3 + 17u4 + 39u5 + 100u6 + 218u7 + 467u8 +865u9 + 1586u10 + 2685u11 + 4467u12 + 6889u13 + 10423u14 +14934u15 + 20921u16 + 27849u17 + 36293u18 + 45278u19 + 55254u20 +64697u21 + 74134u22 + 81782u23 + 88328u24 + 91866u25 + 93539u26 +91866u27 + 88328u28 + 81782u29 + 74134u30 + 64697u31 + 55254u32 +45278u33 + 36293u34 + 27849u35 + 20921u36 + 14934u37 + 10423u38 +6889u39 + 4467u40 + 2685u41 + 1586u42 + 865u43 + 467u44 + 218u45 +100u46 + 39u47 + 17u48 + 5u49 + u50 + u51 + u52 . (4.37) HSI1,4 (z) =

N I1,4 (z) , DI1,4 (z)

(4.38)

where DI1,4 (z) = (1 − z 3 )(1 − z 4 )3 (1 − z 5 )2 (1 − z 6 )2 (1 − z 7 )(1 − z 8 )2 , N I1,4 (z) = 1 + z + z 2 + 3z 3 + 8z 4 + 15z 5 + 32z 6 + 67z 7 + 129z 8 + 217z 9 +355z 10 + 546z 11 + 812z 12 + 1122z 13 + 1511z 14 + 1948z 15 + 2447z 16 +2923z 17 + 3410z 18 + 3827z 19 + 4183z 20 + 4375z 21 + 4461z 22 + 4375z 23 +4183z 24 + 3827z 25 + 3410z 26 + 2923z 27 + 2447z 28 + 1948z 29 + 1511z 30 +1122z 31 + 812z 32 + 546z 33 + 355z 34 + 217z 35 + 129z 36 + 67z 37 + 32z 38 +15z 39 + 8z 40 + 3z 41 + z 42 + z 43 + z 44 . (4.39) Theorem 4.9. Krull dimensions of Sibirsky algebras S1,4 and SI1,4 are equal to (S1,4 ) = 13 and (SI1,4 ) = 11, respectively.

72

The Center and Focus Problem

Hilbert Series for Sibirsky Algebras S1,5 , SI1,5 and Their Krull Dimensions

4.6

From (3.18)–(3.19) for Γ = {1, 5}, by accepting  = 1 and m0 = 1, m1 = 5 and introducing for convenience the notation z0 = b, z1 = f , we find for comitants of the differential system s(1, 5) the initial form of the generating function ϕ1,5 (u) = (1 − u−2 )ψ1 (u)ψ5 (u), (0)

(0)

(0)

(4.40)

where 1 , − b)2 (1 − u−2 b) (1 − 1 (0) ψ5 (u) = (1 − u6 f )(1 − u4 f )2 (1 − u2 f )(1 − f 2 )2 (1 − u−2 f )2 1 · . −4 (1 − u f )2 (1 − u−6 f ) (0)

ψ1 (u) =

u2 b)(1

(4.41)

(0)

Using Sylvester’s advanced method of decomposition of the function ϕ1,5 (u) to elementary fractions, by analogy with Example 3.5 and taking into account Cayley functional equation (3.21) and Conclusion 3.1, we obtain that the following statement is true Theorem 4.10. Ordinary Hilbert series H(S1,5 , u, b, f ) for Sibirsky algebra S1,5 of system s(1, 5) is a rational function on u, b, f and has the form H(S1,5 , u, b, f ) =

N1,5 (u, b, f ) , D1,5 (u, b, f )

(4.42)

where D1,5 (u, b, f ) = (1 − b)(1 − b2 )(1 − bu2 )(1 − bf )2 (1 − b2 f )2 (1 − b3 f )(1 + f ) ·(1 − f 2 )2 (1 − f 3 )3 (1 − f 4 )2 (1 − f 5 )2 (1 − f u2 )2 (1 − f u4 )2 (1 − f u6 ), (4.43) 8  R2k (b, f )u2k , N1,5 (u, b, f ) = k=0

and R2k (b, f ) is from Appendix 2. According to Observation 3.5 from Theorem 4.10, we have Consequence 4.5. A generalized Hilbert series H(SI1,5 , b, f ) of Sibirsky algebra SI1,5 has the form H(SI1,5 , b, f ) =

N I1,5 (b, f ) , DI1,5 (b, f )

(4.44)

Hilbert Series for Sibirsky Algebras

73

where DI1,5 (b, f ) = (1 − b)(1 − b2 )(1 − bf )2 (1 − b2 f )2 (1 − b3 f )(1 + f ) ·(1 − f 2 )2 (1 − f 3 )3 (1 − f 4 )2 (1 − f 5 )2 , N I1,5 (b, f ) = R0 (b, f ),

(4.45)

and R0 (b, e) is from Appendix 2. Using equalities (3.52) and (3.54) from expressions (4.42)–(4.45), we obtain that there takes place Theorem 4.11. Ordinary Hilbert series HS1,5 (u) and HSI1,5 (z) for Sibirsky algebras S1,5 and SI1,5 of system s(1, 5) from (1.5) have the form N1,5 (u) , D1,5 (u)

H(S1,5 , u) =

(4.46)

where D1,5 (u) = (1 − u4 )4 (1 − u6 )5 (1 − u8 )4 (1 − u10 )2 ,

(4.47)

N1,5 (u) = 1 + u2 + u4 + 3u6 + 27u8 + 70u10 + 177u12 + 338u14 +644u16 + 1090u18 + 1800u20 + 2640u22 + 3689u24 + 4658u26 +5555u28 + 6063u30 + 6317u32 + 6063u34 + 5555u36 + 4658u38 +3689u

40

+ 2640u

+177u

52

42

+ 70u

+ 1800u

54

+ 27u

44

56

+ 1090u

+ 3u

HSI1,5 (z) =

58

46

+u

+ 644u

60

+u

62

48

+ 338u

(4.48)

50

+ u64 .

N I1,5 (z) , DI1,5 (z)

(4.49)

where DI1,5 (z) = (1 − z 2 )4 (1 − z 3 )4 (1 − z 4 )3 (1 − z 5 )2 , N I1,5 (z) = 1 + z + 9z 4 + 22z 5 + 50z 6 + 79z 7 + 120z 8 + 160z 9 +221z 10 + 269z 11 + 325z 12 + 339z 13 + 325z 14 + 269z 15 +221z

16

+ 160z

17

+ 120z

18

+z

+ 79z

25

19

+ 50z

20

+ 22z

21

+ 9z

(4.50)

22

26

+z .

Theorem 4.12. Krull dimensions of Sibirsky algebras S1,5 and SI1,5 are equal to (S1,5 ) = 15 and (SI1,5 ) = 13, respectively.

74

The Center and Focus Problem

4.7

Obtaining Ordinary Hilbert Series for Sibirsky Algebras S1,2,3 , SI1,2,3 Using, Residues, and Calculating Krull Dimensions for Them

Let G be a linear reductive group on algebraically closed field K and V – an n–dimensional rational representation. Hilbert series of the ring of invariants K[V ]G are denoted by H(K[V ]G , t). [12]. Theorem 4.13. (Molien’s formula [12]). Let G be a finite group acting on a finite-dimensional vector space V over a field K of characteristic, not divisible by |G|. Then H(K[V ]G , t) =

1  1 . 0 |G| detV (1 − tσ) σ∈G

If K has the characteristic 0, then det0V (1 − tσ) can be taken as detV (1 − tσ). Assume that char(K) = 0. From Theorem 4.13, it follows that for a finite group, Hilbert series of the ring of invariants can be easily calculated. If G is a finite group, and V is a finite-dimensional representation, then according to [12], we have 1  1 H(K[V ]G , t) = . (4.51) |G| detV (1 − tσ) σ ∈G

This idea can be generalized to arbitrary reductive groups. Assume that K is a field of complex % numbers C. Then, we can choose the Haar measure dμ on C with the norm dμ = 1. Let V be a finite-dimensional rational representation C

for G. Then, according to [12], the following expression is the generalization of (4.51): & dμ G . (4.52) H(C[V ] , t) = det (1 − tσ) V C Note that the Hilbert series H(C[V ]G , t) converges for |t| < 1, because this series is a rational function with poles only in t = 1. Since C is compact, there exists a constant A > 0 such that for every σ ∈ C and every eigenvalue λ, for σ we have |λ| ≤ A. Since λ is an eigenvalue of σ  , it follows that |λ | ≤ A for all . This means that |λ| ≤ 1. Obviously, the integral of the right-hand side of (4.52) is also defined for |t| < 1 [12]. Assume that G is a connected group. Let T be a maximal torus for G, and D the maximal compact subgroup for T . We may assume that C contains D. The torus can be identified with (C∗ )r , where r is the rank of G and D can be identified with the subgroup (S 1 )r of (C∗ )r , where C∗ ⊃ S 1 is a unit % circle. We can choose a Haar measure dν on D such that the equality dν = 1 D

holds. % Suppose that f is a continuous class of functions on C. An integral like f (σ)dμ can be considered as an integral over D, since f is constant on C

Hilbert Series for Sibirsky Algebras

75

conjugacy classes. More precisely, there exists a weight function ϕ : D → R, such that for every continuous class of functions, f we have [12] & & f (σ)dμ = ϕ(σ)f (σ)dν. C

D

Then we obtain [12] & G

H(C[V ] , t) = C

dμ = detV (1 − tσ)

& D

ϕ(σ)dν . detV (1 − tσ)

(4.53)

Choosing the appropriate bases in V and its adjoint space V ∗ , we can achieve that the compact torus D acts diagonally on V and V ∗ . Then the action (z1 , ..., zr ) ∈ D on V ∗ is defined by the matrix ⎛ ⎞ m1 (z) 0 ... 0 ⎜ ⎟ 0 m2 (z) . . . 0 ⎜ ⎟ , ⎜ ⎟ .. .. . .. .. ⎠ ⎝ . . . 0

0

...

mn (z)

where m1 , m2 , ..., mn are Laurent monomials of z1 , ..., zr . In these notations, we have detV (1 − t · (z1 , ..., zn )) = (1 − m1 (z)t)· (1 − m2 (z)t) · · · (1 − mn (z)t). Hence, according to [12], we obtain & ϕ(z)dν . (4.54) H(C[V ]G , t) = − − (1 m (z)t)(1 m2 (z)t) · · · (1 − mn (z)t) 1 D It follows that H(K[V ]G , t) is the coefficient z ρ (in the form of series in z1 , ..., zr with coefficients at K(t)) in the expression  w(ρ) w∈W sgn(w)z , (1 − m1 (z)t)(1 − m2 (z)t) · · · (1 − mn (z)t) or coefficient z 0 = 1 in

 z −ρ w∈W sgn(w)z w(ρ) . (1 − m1 (z)t)(1 − m2 (z)t) · · · (1 − mn (z)t)

Recall Residue Theorem from the theory of functions of a complex variable, which can be applied to calculate Hilbert series for rings of invariants. Suppose that f (z) is a meromorphic function on C. If a ∈ C, then f can be written as a Laurent series: f (z) =

∞  k=−d

at the point z = a.

ck (z − a)k

76

The Center and Focus Problem

If d > 0 and c−d = 0, then f has a pole in z = a of order d. Residue of the function f in z = a is denoted by Res(f, a) and is determined by the equality Res(f, a) = c−1 . If the order k of the pole z = a of the function f satisfies the inequality k ≥ 1, then residue can be calculated from the formula Res(f, a) =

1 dk−1 lim k−1 ((z − a)k f (z)). (k − 1)! z→a dz

Choose D such that γ : [0, 1] → C is a smooth curve. Then the integral over the curve γ is determined by the equality &

&

1

f (z)dz = γ

f (γ(t))γ  (t)dt.

0

Theorem 4.14. (Residue Theorem [12]). Suppose that D is a closed, simply connected compact domain in C, whose border is ∂D, and γ : [0, 1] → C is such a smooth curve, that γ([0, 1]) = ∂D, γ(0) = γ(1), surrounding D exactly once counterclockwise. Suppose that f is a meromorphic function on C without poles in ∂D. Then we have the formula &  1 f (z)dz = Res(f, a). 2πi γ a∈D

Note that in D there is a finite number of points at which f has a nonzero residue. Example 4.1 [12]. Let T = Gm is a one-dimensional torus acting on a three-dimensional space V with the matrix ⎛ ⎞ z 0 0 0 ⎠. =⎝ 0 z −2 0 0 z The action Gm on V ∗ is given by the matrix ⎛ −1 ⎞ z 0 0 ⎝ 0 z −1 0 ⎠ . 0 0 z2 Thus, we obtain the equality HT (K[V ], z, t) =

1 . (1 − z −1 t)2 (1 − z 2 t)

(4.55)

Hilbert Series for Sibirsky Algebras

77

In order that the Hilbert series converge, it is necessary that |z −1 t| < 1 and |z 2 t| < 1. Suppose that |z| = 1 and |t| < 1. To find the coefficient of z 0 , divide (4.55) by 2πiz and integrate on the unit circle S 1 in C. Then we have & dz 1 . (4.56) H(K[V ]Gm , t) = −1 2πi S 1 z(1 − z t)2 (1 − z 2 t) According to Residue Theorem 4.14, the expression from (4.56) can be written as &  1 f (z)dz = Res(f (z), a), 2πi S 1 1 a∈D

where D1 is a unit circle and f (z) = z −1 (1 − z −1 t)−2 (1 − z 2 t)−1 . The poles of the function f (z) are only z = t and z = ±t−1/2 . Since |t| < 1, then the only pole in the unit circle is z = t. Calculating the residue, we have 1 1 z = . (z − t)2 1 − z 2 t z(1 − z −1 t)2 (1 − z 2 t) Power series for the g(z) =

z 1−z 2 t

in a neighborhood of z = t is given by

g(z) = g(t) + g  (t)(z − t) + =

g  (t)(z − t)2 + ... 2

t 1 + t3 + (z − t) + ... . (1 − t3 )2 1 − t3

(4.57)

The Hilbert series H(K[V ]Gm , t) is a residue of the function 1 g(z) = z(1 − z −1 t)2 (1 − z 2 t) (z − t)2 in z = t. Then, from (4.57) it follows that the indicated series has the form 1 + t3 . (1 − t3 )2 From [11] it is known Theorem 4.15. H(K[V ]G , t) =

1 2πi

& S1

1 dz det(I − tρV (z)) z

where S 1 ⊂ C is a unit circle {z : |z| = 1}. Using Residue Theorem and the corresponding generating function (3.18)– (3.19), we conclude that the last theorem can be adapted to calculate an

78

The Center and Focus Problem

ordinary Hilbert series for algebra of comitants and invariants of differential systems. Theorem 4.16. 1 HSIΓ (t) = 2πi

& S1

(0)

ϕΓ (z) dz, z (0)

where S 1 ⊂ C is a unit circle {z : |z| = 1}, ϕΓ (z) – corresponding generating function (3.18)–(3.19). Using Theorem 4.16, the following statement is obtained Theorem 4.17. For a cubic differential system s(1, 2, 3), the ordinary Hilbert series for Sibirsky algebras S(1, 2, 3) and SI(1, 2, 3) of comitants and invariants are as follows: HS1,2,3 (t) =

1 (1 − t (1 − t)2 (1 − t2 )2 (1 − t3 )6 (1 − t4 )3 (1 − t5 )3 (1 − t7 )

+3t2 + 9t3 + 36t4 + 90t5 + 220t6 + 459t7 + 946t8 + 1748t9 +3032t10 + 4845t11 + 7302t12 + 10268t13 + 13749t14 +17327t15 + 20781t16 + 23565t17 + 25460t18 + 26051t19 +25460t20 + 23565t21 + 20781t22 + 17327t23 + 13749t24 +10268t25 + 7302t26 + 4845t27 + 3032t28 + 1748t29 +946t30 + 459t31 + 220t32 + 90t33 + 36t34 + 9t35 +3t36 − t37 + t38 ), 1 HSI1,2,3 (t) = (1 (1 − t)(1 − t2 )3 (1 − t3 )5 (1 − t4 )2 (1 − t5 )3 (1 − t7 ) +t2 + 6t3 + 24t4 + 57t5 + 128t6 + 244t7 + 447t8 + 756t9 + 1203t10 +1760t11 + 2433t12 + 3124t13 + 3800t14 + 4351t15 + 4736t16 +4854t17 + 4736t18 + 4351t19 + 3800t20 + 3124t21 + 2433t22 +1760t23 + 1203t24 + 756t25 + 447t26 + 244t27 +128t28 + 57t29 + 24t30 + 6t31 + t32 + t34 ). From this theorem it follows that a Krull dimension of the Sibirsky graded algebra S1,2,3 (SI1,2,3 ) is equal to 17 (15). We note that the method of calculating the ordinary Hilbert series using Theorem 4.16 for Sibirsky algebras of different differential systems was confirmed on the following Hilbert series: HSI1 , HS2 , HSI2 , HSI0,2 , HSI1,2 , HSI1,3 , HSI2,3 , HS5 , HSI5 , known from [33,34]. Remark 4.4. Note that for Hilbert series of Sibirsky graded algebra of comitants of the system s(Γ), where Γ  0, the following equality holds: HSΓ (t) = HSIΓ∪{0} (t).

Hilbert Series for Sibirsky Algebras

4.8

79

Comments to Chapter Four

As it follows from the papers [33,34], the generalized and the ordinary Hilbert series for Sibirsky algebras of differential systems of the form (1.1)–(1.2) play an important role in studying the structures of these algebras. With the help of these series, one can get an idea of the upper bound of degrees of generators of these algebras. It is known from [40] that if the ordinary Hilbert series for (t) commutative algebras has the form H(t) = N D(t) , where D(t) and N (t) are polynomials in t, then degD(t) > degN (t). In our case, for Sibirsky algebras, this inequality also holds. If these algebras are written in the form (4.1) and NA (t) their ordinary Hilbert series in the form HA (t) = D , then from all examples A (t) concerning the construction of the generators of algebras A at the Chi¸sin˘au school of differential equations, it was found that deg ai ≤ degDA (t), where 1≤i≤m

deg ai denotes the degree of generators of these algebras with respect to the coefficients and phase variables of corresponding differential systems. In addition, the ordinary Hilbert series for Sibirsky algebra makes it possible to calculate the Krull dimension of the given algebra, and at the same time, to estimate the upper bound of these dimensions for their subalgebras for which these series are unknown.

5 About the Center and Focus Problem

5.1

On a New Formulation of the Center and Focus Problem for Differential Systems s(1, m1 , m2 , ..., m` )

Consider the system s(1, m1 , m2 , ..., m` ) of the form (1.1)–(1.2). Suppose that the roots of characteristic equation of the considered system are purely imaginary, i.e., a singular point O(0, 0) of this system is a center for it (surrounded by closed trajectories) or a focus (surrounded by spirals) [2], [20]. The center and focus problem can be formulated as follows: Let for the system s(1, m1 , m2 , ..., m` ) from (1.1)–(1.2) the origin of coordinates be a singular point of the second type (center or focus), then what will be the conditions which distinguish a center from a focus? This problem was posed by H. Poincar´e [26]. Fundamental results were obtained by A. Lyapunov [20] which showed that the conditions for center are the vanishing of an infinite sequence of polynomials (focus quantities) L1 , L2 , ..., Lk , ...

(5.1)

in the coefficients of right side of the system s(1, m1 , ..., m` ) of the form (1.1)– (1.2). If at least one of the quantities from (5.1) is nonzero, then the origin of coordinates for the system s(1, m1 , ..., m` ) of the form (1.1)–(1.2) is a focus. These conditions are necessary and sufficient. In the case of the system s(1, m1 , ..., m` ) from Hilbert’s theorem on the finiteness of basis of polynomial ideals, it follows that essential conditions for center in the indicated sequence are only a finite number, and the rest are consequences of them. Then the center and focus problem for the system s(1, m1 , ..., m` ) of the form (1.1)–(1.2) takes the following formulation: how many polynomials ω (essential conditions for center) Ln1 , Ln2 , ..., Lnω , ... (ni ∈ {1, 2, ..., k, ...}; i = 1, ω; ω < ∞)

(5.2)

from (5.1) is it necessary to attract so that their equality to zero annuls all other polynomials in (5.1)? In the works of academician K. S. Sibirsky [37–39] and his disciples [6]–[8], [10], it was shown that for some systems of the form (1.1)–(1.2), essential conditions of center (5.2) are expressed through centro-affine comitants of these DOI: 10.1201/9781003193074-6

81

82

The Center and Focus Problem

systems. This prompted the authors not to look for an explicit form of center conditions but to determine the relationship between the number of focus quantities (5.2) and some characteristics of the set of centro-affine comitants and invariants. From here arose the generalized center and focus problem in the following formulation: to determine the upper bound of number of algebraically independent essential focus quantities involved in solving the center and focus problem for any system s(1, m1 , ..., m ) of the form (1.1)–(1.2).

5.2

Sibirsky Invariant Variety for Center and Focus

The center and focus problem for differential systems s(1, m1 , ..., m ) of the form (1.1)–(1.2) has the following classical formulation: for an infinite system of polynomials (5.3) {(x2 + y 2 )k }∞ k=1 there exists such function U (x, y) = x2 + y 2 +

∞ 

fk (x, y),

(5.4)

k=3

where fk (x, y) are homogeneous polynomials of degree k with respect to variables x, y, and such constants L1 , L2 , ..., Lk , ...,

(27.1)

that the following identity holds: ∞

 dU = Lk (x2 + y 2 )k+1 dt

(5.5)

k=1

(with respect to x and y) along the trajectories of the Lyapunov system [20] x˙ = y +

 

Pmi (x, y), y˙ = −x +

i=1

 

Qmi (x, y),

(5.6)

i=1

which we denote by sL(1, m1 , ..., m ). As it follows from the theory of invariants and comitants of differential systems [33,37,43], the algebra S1,m1 ,...,m for any differential system s(1, m1 , ..., m ), written as x˙ = cx + dy +

  i=1

Pmi (x, y), y˙ = ex + f y +

  i=1

Qmi (x, y)

(5.7)

About the Center and Focus Problem

83

contains, among its generators, the polynomials i1 = c + f, i2 = c2 + 2de + f 2 , k2 = −ex2 + (c − f )xy + dy 2 ,

(5.8)

given earlier in (2.12). Remark 5.1. Note that the set V = {i1 = c + f = 0, Discr(k2 ) = 2i2 − i21 < 0}

(5.9)

will be called the Sibirsky invariant variety of center and focus for differential system (5.7), since the comitant k2 from (5.8), using a real centro-affine transformation of the plane xOy on the variety V , can be reduced to the form x2 + y 2 ,

(5.10)

and system (5.7) (scalar change of time t is allowed here) can be reduced to the form (5.6) [37], for which the roots of the characteristic equation are purely imaginary, i.e., the origin of coordinates for this system is a singular point of the second group (center or focus). According to Remark 5.1, we obtain Observation 5.1. If we consider the expression of the comitant k2 from (5.8) and the fact that with the help of a real centro-affine transformation on Sibirsky invariant variety V its expression can be reduced to the form (5.10), then formally this variety for differential system (5.7) can be written as V = {f = −c} ∪ {c = 0, d = −e = 1}.

5.3

(5.11)

Focus Quantities Lk and Constants Gk on Sibirsky Invariant Variety of the System s(1, m1 , ..., m ) and Null Focus Pseudo-Quantity

Consider the center and focus problem for differential system (5.7). Then, for this system, we write the identity



   ∞   ∂U ∂U  cx+dy + Pmi (x, y) + ex+f y + Qmi (x, y) = Gk k2k+1 , ∂x ∂y i=1 i=1 k=1 (5.12) where ∞  Fr (x, y), U (x, y) = k2 + (5.13) r=3

and k2 ≡ 0 from (5.8) and Fr (x, y) are homogeneous polynomials of degree r with respect to x, y. Equality (5.12) splits by powers of x and y into an

84

The Center and Focus Problem

infinite number of algebraic equations, where the quantities G1 , G2 , ..., Gk , ..., are variables as well as coefficients of the polynomials Fr (x, y). For any differential system (5.7) from identity (5.12) with k2 from (5.8), we find that the first three equations have the form x2 : e(c + f ) = 0, xy : (c − f )(c + f ) = 0, y 2 : d(c + f ) = 0. These equations are equivalent to one of two series of the conditions: 1) c + f = 0; 2) e = c − f = d = 0. Since k2 ≡ 0, then, according to (5.8), these conditions are equivalent with the first equality c + f = 0, which is contained in the variety V from (5.9). Following the abovementioned, according to Remark 5.1 and the formulated center and focus problem for differential system (5.6), we conclude that for focus quantities Lk from (5.1) and constants Gk from (5.12) the following equalities hold: Lk = Gk |V (k = 1, 2, ...),

(5.14)

where V is from (5.9). From the abovementioned, it follows Observation 5.2. Identity (5.12) with function (5.13) on the variety V from (5.9) allows us to state that differential system (5.7) under these conditions has a singular point of the second group at the origin of coordinates (center or focus). We denote the expression c + f = 0, which is contained in the variety V from (5.9), by G0 ≡ c + f = 0, (5.15) and will call it null focus pseudo-quantity. We note that G0 from (5.15) is a centro-affine (unimodular) invariant of the system s(1, m1 , ..., m ) of a type (0, 1, 0, ..., 0). ' () * 

To obtain a more clear idea identity (5.12) with function the remaining equations from x3 , x2 y, xy 2 , y 3 , ... for various considering equality (5.15).

5.4

about the quantities G1 , G2 , ..., Gk , ... from (5.13), let us study, in the next chapter, the expansion of this identity in powers of differential systems s(1, m1 , ..., m ) without

Polynomials in Coefficients of Differential Systems that Have Weight Isobarity (h, g)

Consider a special case of differential system (5.7) ( = 1, m1 = 2; s(1, 2)) written in a tensor form (this form of writing differential systems of the form

About the Center and Focus Problem

85

(1.1)–(1.2) was introduced by academician K. S. Sibirsky [39] in the 60s of the last century): (5.16) x˙ j = ajα xα + ajαβ xα xβ (j, α, β = 1, 2), where the tensor coefficient ajαβ is symmetric in the lower indices, by which a total convolution is carried out here. Differential system (5.16) in an expanded form is written in the following form: x˙ 1 = a11 x1 + a12 x2 + a111 (x1 )2 + 2a112 x1 x2 + a122 (x2 )2 , (5.17) 2 1 2 x x + a222 (x2 )2 . x˙ 2 = a21 x1 + a22 x2 + a211 (x1 )2 + 2a12 Note that if we introduce the notation x = x1 , c = a11 , d = a12 , g = a111 , h = a112 , k = a122 , 2 2 , n = a22 , y = x2 , e = a21 , f = a22 , l = a211 , m = a12

(5.18)

then we obtain previously encountered system (1.5) and vice versa. From the theory of invariants of differential systems (1.1)–(1.2) [43], the difference between the number of subscripts and superscripts that are equal to 1 (2), will be called as a weight of any coordinate of the tensor coefficient ajα or ajαβ relative to the coordinate x1 (x2 ). For example: (1) Weight of a12 relative to x1 is equal to −1, and relative to x2 is equal to 1; (2) Weight of a11 relative to x1 is equal to 0, and relative to x2 is equal to 0; (3) Weight of a12 relative to x1 is equal to 1, and relative to x2 is equal to −1; (4) Weight of a111 relative to x1 is equal to 1, and relative to x2 is equal to 0; (5) Weight of a112 relative to x1 is equal to 0, and relative to x2 is equal to 1 etc. If the polynomial S is considered, which depends on coefficients of differential system (5.17) of the form (0, d0 , d1 ), (5.19) i.e. of homogeneous degree d0 relative to coordinates of the tensor ajα and of degree d1 relative to coordinates of the tensor ajαβ , then the weight of each member of this polynomial relatively to x1 or x2 is equal to the sum of the weights corresponding to each factor of this member with respect to x1 or x2 . Zero in (5.19) indicates that the expression S does not contain the phase variables x1 , x2 . For example, the monomial a11 a21 a122 has type (0, 2, 1), and weight is equal to 0 relative to x1 , and weight is equal to 1 relative to x2 .

86

The Center and Focus Problem

If all members of the polynomial S of type (5.19) have the weight h relative to x1 and the weight g relative to x2 , then we say that the polynomial S has weight isobarity (h, g) [43]. The isobaric property of polynomials is of great importance in the theory of invariants. For example, if we want to check if any polynomial S in coefficients of system (5.17) can be a coefficient in some comitant of type (δ, d0 , d1 ) of this system, it is necessary that the polynomial S has isobarity of some weight (h, g). From the work [37], it is known that a comitant K11 of the system (5.17) has type (3, 1, 1) and the form K11 = A0 (x1 )3 + A1 (x1 )2 x2 + A2 x1 (x2 )2 + A3 (x3 )3 ,

(5.20)

where 1 2 − 2a22 a12 , A0 = −a12 a111 − a22 a211 , A1 = a11 a111 + a21 a211 − 2a21 a12 2 , A3 = a11 a122 + a12 a222 . A2 = 2a11 a112 + 2a12 a212 − a21 a122 − a22 a22

(5.21)

Note that in A0 all terms have weight isobarity (2, −1), in A1 –of weight (1, 0), in A2 –of weight (0, 1) and in A3 –of weight (−1, 2). In the above example, the expression A0 from (5.21) is a semi-invariant of the comitant K11 from (5.20). Further, we will see that semi-invariant of any comitant has a special meaning. Note that for the semi-invariant of weight isobarity (h, g) in any comitant, the number g coincides with the weight of this comitant.

5.5

Comments to Chapter Five

In this chapter, we formulate the generalized center and focus problem and the arguments that prompted the authors to such formulation. We consider the Sibirsky invariant variety of center and focus, which is closely related to centro-affine classification of the quadratic form (comitant) k2 (see [37], p. 31) of the system s(1, m1 , ..., m ). The concepts of isobarity and semi-invariants, which play an important role in the construction of centro-affine comitants and invariants, are explained.

6 On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the System s(1, m1, ..., m`)

6.1

Applications of Generating Functions and Hilbert Series to the Center and Focus Problem for the Differential System s(1, 2)

Consider the differential system s(1, 2), which we write as follows: x˙ = cx + dy + gx2 + 2hxy + ky 2 , y˙ = ex + f y + lx2 + 2mxy + ny 2

(6.1)

with a finitely defined graded algebra of unimodular comitants S1,2 [33]. For this system, we write function (5.13) in the form U = k2 + a0 x3 + 3a1 x2 y + 3a2 xy 2 + a3 y 3 + b0 x4 + 4b1 x3 y + 6b2 x2 y 2 +4b3 xy 3 + b4 y 4 + c0 x5 + 5c1 x4 y + 10c2 x3 y 2 + 10c3 x2 y 3 + 5c4 xy 4 +c5 y 5 + d0 x6 + 6d1 x5 y + 15d2 x4 y 2 + 20d3 x3 y 3 + 15d4 x2 y 4 + 6d5 xy 5 +d6 y 6 + e0 x7 + 7e1 x6 y + 21e2 x5 y 2 + 35e3 x4 y 3 + 21e5 x2 y 5 + 7e6 xy 6

(6.2)

+e7 y 7 + f0 x8 + 8f1 x7 y + 28f2 x6 y 2 + 56f3 x5 y 3 + 70f4 y 4 +56f5 x3 y 5 + 28f6 x2 y 6 + 8f7 xy 7 + f8 y 8 + ..., where k2 6≡ 0 is from (5.8), and a0 , a1 , ..., f7 , f8 , ... are unknown coefficients. Identity (5.12) along the trajectories of differential system (6.1) with function (6.2) splits into the following systems of equations (equality (5.15) is omitted): x3 : 3ca0 + 3ea1 = 2eg − (c − f )l, x2 y : 3da0 + 3(2c + f )a1 + 6ea2 = (f − c)(g + 2m) − 2dl + 4eh, xy 2 : 6da1 + 3(2f + c)a2 + 3ea3 = (f − c)(2h + n) + 2ek − 4dm,

(6.3)

y 3 : 3da2 + 3f a3 = (f − c)k − 2dn; DOI: 10.1201/9781003193074-7

87

88

The Center and Focus Problem x4 : 4cb0 + 4eb1 − e2 G1 = −3ga0 − 3la1 , x3 y :4db0 + 4(f + 3c)b1 + 12eb2 + 2e(c − f )G1 = −6ha0 − 6(g + m)a1 − 6la2 , x2 y 2 : 12db1 + 12(c + f )b2 + 12eb3 + [2de − (c − f )2 ]G1 = = −3ka0 − 3(4h + n)a1 − 3(g + 4m)a2 − 3la3 , xy 3 : 12bd2 + 4(3f + c)b3 + 4eb4 + 2d(f − c)G1 = −6ka1 − 6(h + n)a2 − 6ma3 , y 4 : 4db3 + 4f b4 − d2 G1 = −3ka2 − 3na3 ;

(6.4)

x : 5cc0 + 5ec1 = −4gb0 − 4lb1 , 5

x4 y : 5dc0 + 5(4c + f )c1 + 20ec2 = −8hb0 − 4(3g + 2m)b1 − 12lb2 , x3 y 2 : 20dc1 + 10(3c + 2f )c2 + 30ec3 = −4kb0 − 4(6h + n)b1 − 12(g + 2m)b2 − 12lb3 , x2 y 3 : 30dc2 + 10(2c + 3f )c3 + 20ec4 = −12kb1 − 12(2h + n)b2 − 4(g + 6m)b3 − 4lb4 , xy 4 : 20dc3 + 5(c + 4f )c4 + 5ec5 = −12kb2 − 4(2h + 3n)b3 − 8mb4 , y 5 : 5dc4 + 5f c5 = −4kb3 − 4nb4 ;

(6.5)

x6 : 6cd0 + 6ed1 + e3 G2 = −5gc0 − 5lc1 , x5 y : 6dd0 + 6(5c + f )d1 + 30ed2 + 3e2 (f − c)G2 = −10hc0 − 10(2g + m)c1 − 20lc2 , x4 y 2 : 30dd1 + 30(2c + f )d2 + 60ed3 + 3e[(c − f )2 − de]G2 = = −5kc0 − 5(8h + n)c1 − 10(3g + 4m)c2 − 30lc3 , x3 y 3 : 60dd2 + 60(c + f )d3 + 60ed4 + (f − c)[(c − f )2 − 6de]G2 = −20kc1 − 20(3h + n)c2 − 20(g + 3m)c3 − 20lc4 , x2 y 4 : 60dd3 + 30(c + 2f )d4 + 30ed5 + 3d[de − (c − f )2 ]G2 = = −30kc2 − 10(4h + 3n)c3 − 5(g + 8m)c4 − 5lc5 , xy 5 : 30dd4 + 6(c + 5f )d5 + 6ed6 + 3d2 (f − c)G2 = −20kc3 − 10(h + 2n)c4 − 10mc5 , y 6 : 6dd5 + 6f d6 − d3 G2 = −5kc4 − 5nc5 ; x7 : 7ce0 + 7ee1 = −6gd0 − 6ld1 , x6 y : 7de0 + 7(6c + f )e1 + 42ee2 = −12hd0 − 6(5g + 2m)d1 − 30ld2 ,

(6.6)

Algebraically Independent Focus Quantities

89

x5 y 2 : 42de1 + 7(15c + 6f )e2 + 105ee3 = −6kd0 − 6(10h + n)d1 − 60(g + m)d2 − 60ld3 , x4 y 3 : 105de2 + 5(28c + 21f )e3 + 140ee4 = −30kd1 − 30(4h + n)d2 − 60(g + 2m)d3 − 60ld4 , x3 y 4 : 140de3 + 35(3c + 4f )e4 + 105ee5 = −60kd2 − 60(2h + n)d3 − 30(g + 4m)d4 − 30ld5 , x2 y 5 : 105de4 + 7(6c + 15f )e5 + 42ee6 = −60kd3 − 60(h + n)d4 − 6(g + 10m)d5 − 6ld6 , xy 6 : 42de5 + 7(c + 6f )e6 + 7ee7 = −30kd4 − 6(2h + 5n)d5 − 12md6 , y 7 : 7de6 + 7f e7 = −6kd5 − 6nd6 ;

(6.7)

x8 : 8cf0 + 8ef1 − e4 G3 = −7ge0 − 7le1 , x7 y : 8df0 + 8(7c + f )f1 + 56ef2 + 4e3 (c − f )G3 = = −14he0 − 14(3g + m)e1 − 42le2 , x6 y 2 : 56df1 + 56(3c + f )f2 + 168ef3 + 2e2 [2de − 3(c − f )2 ]G3 = = −7ke0 − 7(12h + n)e1 − 21(5g + 4m)e2 − 105le3 , x5 y 3 : 168df2 + 56(5c + 3f )f3 + 280ef4 + 4e(f − c)[3de − (c − f )2 ]G3 = −42ke1 − 42(5h + n)e2 − 70(2g + 3m)e3 − 140le4 , x4 y 4 : 280df3 + 280(c + f )f4 + 280ef5 + [12de(c − f )2 − 6d2 e2 − (c − f )4 ]G3 = −105ke2 − 35(8h + 3n)e3 − 35(3g + 8m)e4 − 105le5 , x3 y 5 : 280df4 + 56(3c + 5f )f5 + 168ef6 + 4d(f − c)[(c − f )2 − 3de]G3 = −140ke3 − 70(3h + 2n)e4 − 42(g + 5m)e5 − 42le6 , x2 y 6 : 168df5 + 56(c + 3f )f6 + 56ef7 + 2d2 [2de − 3(c − f )2 ]G3 = = −105ke4 − 21(4h + 5n)e5 − 7(g + 12m)e6 − 7le7 , xy 7 : 56df6 + 8(c + 7f )f7 + 8ef8 + 4d3 (f − c)G3 = −42ke5 − 14(h + 3n)e6 − 14me7 , y 8 : 78df7 + 8f f8 − d4 G3 = −7ke6 − 7ne7 .

(6.8)

It is evident that linear systems of equations (6.3)–(6.8) in the variables a0 , a1 , a2 , a3 , b0 , b1 ,...,b4 , c0 , c1 ,...,c5 , d0 , d1 ,...,d6 , e0 , e1 ,...,e7 , f0 , f1 ,...,f8 ,..., G1 , G2 , G3 , ... can be extended by adding, after the last equation from (6.8), an infinite number of equations obtained from the equality of coefficients of xα y β for α + β > 8 in identity (5.12).

90

The Center and Focus Problem

For a clearer reflection of the process of obtaining G1 , we write systems (6.3), (6.4) in the matrix form A1 B1 = C1 ,

(6.9)

where ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ A1 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

3c 3d 0 0 3g 6h 3k 0 0

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ B1 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

3e 3(2c + f ) 6d 0 3l 6(g + m) 3(4h + n) 6k 0

a0 a1 a2 a3 b0 b1 b2 b3 b4 G1

0 0 6e 0 3(2c + f ) 3e 3d 3f 0 0 6l 0 3(g + 4m) 3l 6(h + n) 6m 3k 3n

0 0 0 0 0 0 12e 4(3f + c) 4d ⎞ ⎛ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ , C1 = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝

0 0 0 0 0 0 0 4l 4f

0 0 0 0 4c 4d 0 0 0

0 0 0 0 4e 4(f + 3c) 12d 0 0

0 0 0 0 −e2 2e(c − f ) 2de − (c − f )2 2d(f − c) −d2

0 0 0 0 0 12e 12(c + f ) 12d 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

2eg + (f − c)l (f − c)(g + 2m) − 2dl + 4eh (f − c)(2h + n) + 3ek − 4dm (f − c)k − 2dn 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(6.10)

Since the dimension of the matrix A1 is 9 × 10, then it is clear that we have at least one free variable. Therefore, choosing as a free variable, one of bi (i ∈ {0, ..., 4}) and using the Cramer’s rule for system (6.9), for each fixed i, we obtain G1,i + B1,i bi G1 = , (6.11) σ1,i

Algebraically Independent Focus Quantities

91

where G1,i , B1,i , σ1,i (see, Appendix 3) are polynomials in coefficients of system (6.1), and bi are undetermined coefficients of the function U (x, y) from (6.2). In the future, we will need an explicit form of the operators X1 , ..., X4 of the Lie algebra L4 for system (6.1), expressions of which are obtained from Section 1.5: X1 = x

∂ ∂ ∂ ∂ + D1 , X2 = y + D2 , X3 = x + D3 , X4 = y + D4 , ∂x ∂x ∂y ∂y (6.12)

where

∂ ∂ ∂ ∂ ∂ ∂ −e −g +k − 2l − m , ∂d ∂e ∂g ∂k ∂l ∂m ∂ ∂ ∂ ∂ ∂ D2 = e + (f − c) −e +l + (m − g ) ∂c ∂d ∂f ∂g ∂h ∂ ∂ ∂ +(n − 2h) −l − 2m , ∂k ∂m ∂n (6.13) ∂ ∂ ∂ ∂ ∂ +d − 2h −k D3 = −d + (c − f ) ∂c ∂e ∂f ∂g ∂h ∂ ∂ ∂ +(g − 2m) + (h − n) +k , ∂l ∂m ∂n ∂ ∂ ∂ ∂ ∂ ∂ D4 = −d +e −h − 2k +l −n . ∂d ∂e ∂h ∂k ∂l ∂n By studying matrices (6.10) of system (6.9), we conclude that G1,i from (6.11) are homogeneous polynomials of degree 8 with respect to the linear part, and of degree 2 with respect to the quadratic part of system (6.1). Note that G1,i from (6.11) for i = 0, 1, 2, 3, 4 are homogeneous polynomials of isobarities with weights, respectively (see, Appendix 3): D1 = d

(3, −1), (2, 0), (1, 1), (0, 2), (−1, 3).

(6.14)

According to formula (3.3) for differential system (6.1) and theory of invariants of differential systems [37,43], it follows that the numerators of fractions (6.11) can be coefficients in comitants of the weight −1 of type (4, 8, 2). This means that according to (2.56) with the help of Lie differential operator D3 from (6.13) for system (6.1) and numerator of fraction (6.11), we obtain a system of four linear nonhomogeneous partial differential equations D3 (G1,0 + B1,0 b0 ) = G1,1 + B1,1 b1 , D3 (G1,1 + B1,1 b1 ) = −G1,2 − B1,2 b2 , −D3 (G1,2 + B1,2 b2 ) = G1,3 + B1,3 b3 , D3 (G1,3 + B1,3 b3 ) = −G1,4 − B1,4 b4 , (6.15) with five unknowns b0 , b1 , b2 , b3 , b4 . According to Lemma 2.4, system (6.15) has an infinite number of solutions. Note that a particular solution of this system is b0 = b1 = b2 = b3 = b4 = 0, for which the polynomial f4 (x, y) = G1,0 x4 + 4G1,1 x3 y + 2G1,2 x2 y 2 + 4G1,3 xy 3 + G1,4 y 4

(6.16)

92

The Center and Focus Problem

is a centro-affine comitant of differential system (6.1). This fact is also confirmed by Theorem 2.2 with the operators X1 − X4 from (6.12)–(6.13) for differential system (6.1), for which we have the equalities X1 (f4 ) = X4 (f4 ) = f4 , X2 (f4 ) = X3 (f4 ) = 0. Another particular solution for system (6.15) is given by the following expressions −e(g 2 + 2hl + m2 ) , 3c2 − 4de + 10cf + 3f 2 (c − f )(g 2 + 2hl + m2 ) − 2e(gh + kl + hm + mn) , 4(3c2 − 4de + 10cf + 3f 2 ) 2(c − f )(gh + kl + hm + mn) − e(h2 + 2km + n2 ) + d(g 2 + 2hl + m2 ) , 6(3c2 − 4de + 10cf + 3f 2 ) (c − f )(h2 + 2km + n2 ) + 2d(gh + kl + hm + mn) , 4(3c2 − 4de + 10cf + 3f 2 ) d(h2 + 2km + n2 ) , 3c2 − 4de + 10cf + 3f 2

b0 = b1 = b2 = b3 = b4 =

whose denominators are different from zero on the invariant variety V from (5.11). This solution determines the centro-affine comitant f4 (x, y) = (G1,0 + B1,0 b0 )x4 + 4(G1,1 + B1,1 b1 )x3 y + 2(G1,2 +B1,2 b2 )x2 y 2 + 4(G1,3 + B1,3 b3 )xy 3 + (G1,4 + B1,4 b4 )y 4 .

(6.17)

It is evident that differential system (6.15) has infinite number of solutions b0 , b1 , b2 , b3 , b4 , which define a centro-affine comitants of type (4, 8, 2). In view of the abovementioned, comitants (6.16), (6.17) belong to the space (4,8,2)

S1,2

,

(6.18)

of Sibirsky algebra S1,2 . Note that comitants (6.16), (6.17) on the invariant variety V from (5.11) for differential system (6.1) have the following form f4 (x, y)|V = f4 (x, y)|V = −8L1 (x2 + y 2 )2 (G1 |V = −8L1 ),

(6.19)

where

1 [g(l − h) − k(h + n) + m(l + n)] 2 is the first focus quantity of differential system (6.1) on the invariant variety V and coincides with the focus quantity from [35, p. 110] for system (6.1), received after substitution f = −c = 0, d = −e = 1. Similarly to the previous case, for determining the quantity G2 , we write system of equations (6.3)–(6.6) in the matrix form (see, Appendix 4) L1 =

A2 B2 = C2 ,

(6.20)

Algebraically Independent Focus Quantities

93

from where we find G2,i,j + B2,i,j bi + D2,i,j dj , (i = 0, 4, j = 0, 6). G2 = σ2,i,j

(6.21)

By studying matrix equality (6.20), we obtain that degG2,i,j = 24, and using system (6.3)–(6.6), we obtain that G2,i,j from (6.21) has type (0, 20, 4), i.e. G2,i,j are homogeneous polynomials of degree 20 in coefficients of the linear part and of degree 4 in coefficients of the quadratic part of the system s(1, 2) from (6.1). Computing the expressions G2,i,j for each i = 0, 4 and j = 0, 6, we obtain for their isobarities the following Table 6.1 (see, Appendix 4): Table 6.1. Isobarities with weight of polynomials G2,i,j for the system s(1, 2) G 2,i,j b0 b1 b2 b3 b4

d0 (7,−3) (6,−2) (5,−1) (4,0) (3,1)

d1 (6,−2) (5,-1) (4,0) (3,1) (2,2)

d2 (5,−1) (4,0) (3,1) (2,2) (1,3)

d3 d4 (4,0) (3,1) (3,1) (2,2) (2,2) (1,3) (1,3) (0,4) (0,4) (−1,5)

d5 d6 (2,2) (1,3) (1,3) (0,4) (0,4) (−1,5) (−1,5) (−2,6) (−2,6) (−3,7)

Note that for j = 0, 6, we have G2,0,j 1 |V = (38g 3 h + 46gh3 + 71g 2 hk + 46h3 k + 38ghk 2 + 5hk 3 σ2,0,j 24 −38g 3 l + 3gh2 l − 39g 2 kl + 53h2 kl − 15gk 2 l − 32ghl2 + 15hkl2 − 5gl3 +29g 2 hm + 42ghkm + 13hk 2 m − 79g 2 lm − 54h2 lm − 68gklm − 15k 2 lm −37hl2 m − 5l3 m + 6ghm2 + 6hkm2 − 39glm2 − 29klm2 + 2lm3 + 6g 3 n +109gh2 n + 48g 2 kn + 159h2 kn + 33gk 2 n + 5k 3 n − 34ghln + 116hkln −57gl2 n + 15kl2 n − 48g 2 mn − 54h2 mn − 14gkmn + 8k 2 mn − 138hlmn −62l2 mn − 37gm2 n − 27km2 n + 2m3 n + 72ghn2 + 175hkn2 − 72gln2 +63kln2 − 101hmn2 − 119lmn2 − 6gn3 + 62kn3 − 62mn3 ), (6.22) G2,2,j 1 3 3 2 3 2 3 |V = (62g h − 2gh + 95g hk − 2h k + 38ghk + 5hk − 62g 3 l σ2,2,j 24 +27gh2 l − 39g 2 kl + 29h2 kl − 15gk 2 l − 8ghl2 + 15hkl2 − 5gl3 + 53g 2 hm +66ghkm + 13hk 2 m − 127g 2 lm − 6h2 lm − 68gklm − 15k 2 lm − 13hl2 m −5l3 m + 6ghm2 + 6hkm2 − 63glm2 − 29klm2 + 2lm3 + 6g 3 n +61gh2 n + 72g 2 kn + 63h2 kn + 33gk 2 n + 5k 3 n − 10ghln + 68hkln −33gl2 n + 15kl2 n − 72g 2 mn − 6h2 mn + 10gkmn + 8k 2 mn − 66hlmn −38l2 mn − 61gm2 n − 27km2 n + 2m3 n + 72ghn2 + 127hkn2 − 72gln2 +39kln2 − 53hmn2 − 95lmn2 − 6gn3 + 62kn3 − 62mn3 ), (6.23)

94

The Center and Focus Problem 1 G2,4,j |V = (62g 3 h − 2gh3 + 119g 2 hk − 2h3 k + 62ghk 2 + 5hk 3 − 62g 3 l 24 σ2,4,j

+27gh2 l − 63g 2 kl + 29h2 kl − 15gk 2 l − 8ghl2 + 15hkl2 − 5gl3 + 101g 2 hm +138ghkm + 37hk 2 m − 175g 2 lm − 6h2 lm − 116gklm − 15k 2 lm − 13hl2 m −5l3 m + 54ghm2 + 54hkm2 − 159glm2 − 53klm2 − 46lm3 + 6g 3 n +37gh2 n + 72g 2 kn + 39h2 kn + 57gk 2 n + 5k 3 n + 14ghln + 68hkln −33gl2 n + 15kl2 n − 72g 2 mn − 6h2 mn + 34gkmn + 32k 2 mn − 42hlmn −38l2 mn − 109gm2 n − 3km2 n − 46m3 n + 48ghn2 + 79hkn2 − 48gln2 +39kln2 − 29hmn2 − 71lmn2 − 6gn3 + 38kn3 − 38mn3 ) (6.24) G2,1,j G2,3,j and and on the invariant variety V give uncertainties. σ2,3,j σ2,1,j By studying the isobarities of G2,i,j top-down for each line from Table 6.1, according to the theory of invariants of differential systems [37,43], we find that the numerators of fraction (6.21) can be coefficients in a centroaffine comitants with the corresponding weights −3, −2, −1, 0, 1. Using these weights and formula (3.3) for differential system (6.1), as well as the fact that G2,i,j have type (0, 20, 4), we obtain that the mentioned comitants correspond to the types (10, 20, 4), (8, 20, 4), (6, 20, 4), (4, 20, 4), (2, 20, 4).

(6.25)

As the quantity G2 in (5.12) is a coefficient of homogeneity of degree 6 in the phase variables x and y, then it is logical to choose from (6.25) the type (6, 20, 4),

(6.26)

which corresponds to the expression G2,2,j (j = 0, 6) from Table 6.1. This means that according to (2.56), using Lie differential operator D3 from (6.13) for differential system (6.1) and the numerator of fraction (6.21) for the fixed i = 2, we obtain a system of six linear nonhomogeneous partial differential equations: D3 (G2,2,0 + B2,2,0 b0 + D2,2,0 d0 ) = −(G2,2,1 + B2,2,1 b1 + D2,2,1 d1 ), −D3 (G2,2,1 + B2,2,1 b1 + D2,2,1 d1 ) = G2,2,2 + B2,2,2 b2 + D2,2,2 d2 , D3 (G2,2,2 + B2,2,2 b2 + D2,2,2 d2 ) = −(G2,2,3 + B2,2,3 b3 + D2,2,3 d3 ), −D3 (G2,2,3 + B2,2,3 b3 + D2,2,3 d3 ) = G2,2,4 + B2,2,4 b4 + D2,2,4 d4 , D3 (G2,2,4 + B2,2,4 b4 + D2,2,4 d4 ) = −(G2,2,5 + B2,2,5 b5 + D2,2,5 d5 ), −D3 (G2,2,5 + B2,2,5 b5 + D2,2,5 d5 ) = G2,2,6 + B2,2,6 b6 + D2,2,6 d6 ,

(6.27)

with seven unknowns d0 , d1 , ..., d6 , where bi (i = 0, 4) are defined from system (6.15). According to Lemma 2.4, system (6.27) has infinite number of solutions that define comitants of type (6, 20, 4). Note that obtaining explicit solutions of

Algebraically Independent Focus Quantities

95

system (6.27) is a complicated enough procedure. We will show the importance of homogeneities of the expressions G2,2,j from (6.21) in obtaining a focus quantities of differential system (6.1) on the invariant variety of center and focus V from (5.11). According to (6.26), system (6.27) defines centro-affine comitants belonging to the space (6,20,4)

S1,2

(6.28)

of Sibirsky algebra S1,2 . According to (2.56) and (6.27), each comitant, belonging to this space, can be written as f6 (x, y) = (G2,2,0 + B2,2,0 b2 + D2,2,0 d0 )x6 − (G2,2,1 + B2,2,1 b2 1 1 +D2,2,1 d1 )x5 y + (G2,2,2 + B2,2,2 b2 + D2,2,2 d2 )x4 y 2 − (G2,2,3 3! 2! 1 3 3 +B2,2,3 b2 + D2,2,3 d3 )x y + (G2,2,4 + B2,2,4 b2 + D2,2,4 d4 )x2 y 4 4! 1 1 − (G2,2,5 + B2,2,5 b2 + D2,2,5 d5 )xy 5 + (G2,2,6 + B2,2,6 b2 + D2,2,6 d6 )y 6 . 5! 6! Note that on the invariant variety V from (5.11) for differential system (6.1) the expressions G2,2,j (j = 0, 6) get the form G2,2,0 |V = G2,2,2 |V = G2,2,4 |V = G2,2,6 |V = −2304L2 , G2,2,1 |V = G2,2,3 |V = G2,2,5 |V = 0,

(6.29)

where 24L2 = 62g 3 h − 2gh3 + 95g 2 hk − 2h3 k + 38ghk 2 + 5hk 3 − 62g 3 l +27gh2 l − 39g 2 kl + 29h2 kl − 15gk 2 l − 8ghl2 + 15hkl2 − 5gl3 +53g 2 hm + 66ghkm + 13hk 2 m − 127g 2 lm − 6h2 lm − 68gklm −15k 2 lm − 13hl2 m − 5l3 m + 6ghm2 + 6hkm2 − 63glm2 − 29klm2 +2lm3 + 6g 3 n + 61gh2 n + 72g 2 kn + 63h2 kn + 33gk 2 n + 5k 3 n −10ghln + 68hkln − 33gl2 n + 15kl2 n − 72g 2 mn − 6h2 mn +10gkmn + 8k 2 mn − 66hlmn − 38l2 mn − 61gm2 n − 27km2 n +2m3 n + 72ghn2 + 127hkn2 − 72gln2 + 39kln2 − 53hmn2 −95lmn2 − 6gn3 + 62kn3 − 62mn3 is a second focus quantity of system (6.1) on the invariant variety V . For the second focus quantity corresponding to G2,4,j , we obtain G2,4,0 |V = G2,4,2 |V = G2,4,4 |V = G2,4,6 |V = −2304LS2 , G2,4,1 |V = G2,4,3 |V = G2,4,5 |V = 0,

96

The Center and Focus Problem 24LS2 = 62g 3 h − 2gh3 + 119g 2 hk − 2h3 k + 62ghk 2 + 5hk 3 − 62g 3 l

+27gh2 l − 63g 2 kl + 29h2 kl − 15gk 2 l − 8ghl2 + 15hkl2 − 5gl3 + 101g 2 hm +138ghkm + 37hk 2 m − 175g 2 lm − 6h2 lm − 116gklm − 15k 2 lm −13hl2 m − 5l3 m + 54ghm2 + 54hkm2 − 159glm2 − 53klm2 − 46lm3 +6g 3 n + 37gh2 n + 72g 2 kn + 39h2 kn + 57gk 2 n + 5k 3 n + 14ghln +68hkln − 33gl2 n + 15kl2 n − 72g 2 mn − 6h2 mn + 34gkmn + 32k 2 mn −42hlmn − 38l2 mn − 109gm2 n − 3km2 n − 46m3 n + 48ghn2 + 79hkn2 −48gln2 + 39kln2 − 29hmn2 − 71lmn2 − 6gn3 + 38kn3 − 38mn3 . This expression coincides with the focus quantity from [35 p. 110] for system (6.1), received after substitution f = −c = 0, d = −e = 1. Consider determination of G3 of homogeneity of degree 8 with respect to the phase variables x and y in (6.21). Writing system (6.3)–(6.8) in the matrix form A 3 B 3 = C3 , we obtain G3 =

G3,i,j,k + B3,i,j,k bi + D3,i,j,k dj + F3,i,j,k fk σ3,i,j,k

(6.30)

(i = 0, 4; j = 0, 6; k = 0, 8). Similarly to the previous case, we choose a comitant of the weight −1 of the differential system s(1, 2) from (6.1), which contains the expressions G3,2,j,k + B3,2,j,k b2 + D3,2,j,k dj + F3,2,j,k fk (k = 0, 8) as a semi-invariant, and we find that it belongs to the space (8,37,6) (6.31) S1,2 of Sibirsky algebra S1,2 . Lets consider the extension of system (6.3)–(6.8) obtained from identity (5.12) for differential system (6.1) and function (6.2), which contains the quantity Gk , which we write in a matrix form as follows Ak Bk = Ck . We denote by mGk the number of equations and by nGk the number of unknowns of this system. Note that these numbers are written as mGk = 4 + 5 + 6 + 7 + 8 + 9 + · · · + (2k + 2) + (2k + 3), (k = 1, 2, 3, ...), ' () * ' () * ' () * ' () * G1

G2

G3

Gk

nGk = 4 + 6 + 6 + 8 + 8 + 10 + · · · + (2k + 2) + (2k + 4) . ' () * ' () * ' () * ' () * G1

G2

G3

Gk

Hence we obtain mGk = k(2k + 7), nGk = mGk + k > mGk .

(6.32)

Similarly to the previous cases, from this system, we have Gk =

Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik . σk,i1 ,i2 ,...,ik

(6.33)

Algebraically Independent Focus Quantities

97

Now it is important to determine the degree of the polynomial Gk,i1 ,i2 ,...,ik + +Bk,i1 ,i2 ,...,ik bi1 +· · ·+Zk,i1 ,i2 ,...,ik zik in coefficients of differential system (6.1). Note that the degree of nonzero polynomial coefficient of Gi (i = 1, k) in coefficients of system (6.1) in Cramer’s determinant of the order mGk , when the last column corresponding to the quantity Gk is replaced with the column corresponding to free members, forms the following diagram (coefficients of the last quantity Gk have the degree 2 according to the substitution): G1 , G2 , G3 , ..., Gk−1 , Gk . ↓

↓

↓

↓

↓

2 3 4 k 2 Then the degree of the polynomial Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik in coefficients of differential system (6.1), denoted by NGk , will be written as k(k + 1) NGk = mGk − k + + 1, 2 hence according to (6.32), we obtain N Gk =

1 (5k 2 + 13k + 2). 2

(6.34)

It is the degree of homogeneity of Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · +Zk,i1 ,i2 ,...,ik zik in coefficients of linear and quadratic parts of differential system (6.1), which is contained in a polynomial of type (d) = (δ, d0 , d1 ). Since δ = 2(k + 1), and d1 = 2k, then d0 = NGk − 2k. So we obtain that a comitant of the weight −1 of the differential system s(1, 2) from (6.1), containing the semi-invariant Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · +Zk,i1 ,i2 ,...,ik zik which corresponds to the quantity Gk for k = 1, 2, 3, ..., belongs to the type  1 2 2(k + 1), (5k + 9k + 2), 2k , (6.35) 2 where 2(k + 1) is the degree of homogeneity of the comitant in phase variables 1 x, y; (5k 2 + 9k + 2) is the degree of homogeneity of the comitant in coeffi2 cients of linear part, and 2k is the degree of homogeneity of the comitant in coefficients of the quadratic part of the differential system s(1, 2) from (6.1). Hereafter the expressions Gk,i1 ,i2 , ..., ik + Bk,i1 ,i2 , ..., ik bi1 + ··· +Zk,i1 ,i2 ,...,ik zik , which determine comitants of types (6.35), corresponding to the quantity Gk (k = 1, 2, 3, ...), will be called generalized focus pseudoquantities, and comitants of the type (6.35) for k = 1, 2, 3, ... will be called comitants which contain the generalized focus pseudo-quantities Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik as coefficients. 2 1 Note that the spaces S (2(k+1), 2 (5k +9k+2),2k) are generalized records of spaces (6.18), (6.28), (6.31) for k = 1, 2, 3, ... of Sibirsky algebra S1,2 .

98

The Center and Focus Problem According to the paper [33], using Theorem 4.1, there takes place

Theorem 6.1. Dimension of linear space of centro-affine comitants of type (d) = (δ, d0 , d1 ) for the differential system s(1, 2) from (6.1), denoted by (d) dimR V1,2 , is equal to the coefficient of the monomial uδ bd0 cd1 in decomposition of generalized Hilbert series from (4.13)–(4.15) for Sibirsky algebra S1,2 of comitants of the considered system. Consider the subalgebra S1 ,2 ⊂ S1,2 , which we write in the form S1 ,2 =

#

(d )

S1,2 ,

(6.36)

(d) (d )

where by S1,2 the following linear spaces are denoted: (0,0,0)

S1,2

(0,1,0)

= R, S1,2

(2(k+1), 12 (5k2 +9k+2),2k)

, ..., S1,2

, k = 1, 2, ...,

(6.37)

as well as spaces from S1,2 , which contain all kinds of their products. Since the algebra S1 ,2 is a graded subalgebra in a finitely defined algebra S1,2 , then according to Proposition 4.1, we obtain (S1 ,2 ) ≤ (S1,2 ). From this inequality and from the fact that (S1,2 ) = 9 [33], according to Remark 2.3 on semi-invariants and the fact that generalized focus pseudo-quantities are coefficients of some comitants, there takes place Theorem 6.2. Maximal number of algebraically independent generalized focus pseudo-quantities in the center and focus problem for differential system (6.1) does not exceed 9. According to Proposition 4.2, Remark 5.1, and equality (5.14), there follows that the maximal number of algebraically independent focus quantities Lk (k = 1, ∞) cannot exceed the maximal number of algebraically independent generalized focus pseudo-quantities Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik . Hence, according to equalities (5.5), we have Consequence 6.1. Upper bound of the number of algebraically independent focus quantities that take part in solving the center and focus problem for differential system (6.1) does not exceed 9. (d ) Consider the types of spaces S1,2 from (6.37) for d = δ  + d0 + d1 ≤ 60, which are obtained from expansion in a power series of the fraction (1 − b)(1 − 





u4 b8 c2 )(1

1 , − u6 b20 c4 )(1 − u8 b37 c6 ) (d )

(6.38)

where uδ bd0 cd1 shows a type of space S1,2 for (d ) = (δ  , d0 , d1 ). In consideration of these types and the generalized Hilbert series (4.13)–(4.15) of the  algebra S1,2 , we can write expansion of Hilbert series of the algebra S1,2 for d = δ  + d0 + d1 ≤ 60, which has the form

Algebraically Independent Focus Quantities

99

 H(S1,2 , u, b, c) = 1 + b + 2b2 + 2b3 + 3b4 + 3b5 + 4b6 + 4b7 + 5b8 + 5b9

+6b10 + 6b11 + 7b12 + 7b13 + 8b14 + 8b15 + 9b16 + 9b17 + 10b18 + 10b19 +11b20 + 11b21 + 12b22 + 12b23 + 13b24 + 13b25 + 14b26 + 14b27 + 15b28 +15b29 + 16b30 + 16b31 + 17b32 + 17b33 + 18b34 + 18b35 + 19b36 + 19b37 +20b38 + 20b39 + 21b40 + 21b41 + 22b42 + 22b43 + 23b44 + 23b45 + 24b46 +24b47 + 25b48 + 25b49 + 26b50 + 26b51 + 27b52 + 27b53 + 28b54 + 28b55 +29b56 + 29b57 + 30b58 + 30b59 + 31b60 + u4 (68b8 c2 + 79b9 c2 + 87b10 c2 +98b11 c2 + 106b12 c2 + 117b13 c2 + 125b14 c2 + 136b15 c2 + 144b16 c2 +155b17 c2 + 163b18 c2 + 174b19 c2 + 182b20 c2 + 193b21 c2 + 201b22 c2 +212b23 c2 + 220b24 c2 231b25 c2 + 239b26 c2 + 250b27 c2 + 258b28 c2 +269b29 c2 + 277b30 c2 + 288b31 c2 + 296b32 c2 + 307b33 c2 + 315b34 c2 +326b35 c2 + 334b36 c2 + 345b37 c2 + 353b38 c2 + 364b39 c2 + 372b40 c2 +383b41 c2 + 391b42 c2 + 402b43 c2 + 410b44 c2 + 421b45 c2 + 429b46 c2 +440b47 c2 + 448b48 c2 + 459b49 c2 + 467b50 c2 + 478b51 c2 + 486b52 c2 +497b53 c2 + 505b54 c2 ) + u6 (988b20 c4 + 1046b21 c4 u6 + 1098b22 c4 +1156b23 c4 + 1208b24 c4 + 1266b25 c4 + 1318b26 c4 + 1376b27 c4 + 1428b28 c4 +1486b29 c4 + 1538b30 c4 + 1596b31 c4 + 1648b32 c4 + 1706b33 c4 + 1758b34 c4 +1816b35 c4 + 1868b36 c4 + 1926b37 c4 + 1978b38 c4 + 2036b39 c4 + 2088b40 c4 +2146b41 c4 + 2198b42 c4 + 2256b43 c4 + 2308b44 c4 + 2366b45 c4 + 2418b46 c4 +2476b47 c4 + 2528b48 c4 + 2586b49 c4 + 2638b50 c4 ) + u8 (798b16 c4 +855b17 c4 u8 + 918b18 c4 + 975b19 c4 + 1038b20 c4 + 1095b21 c4 + 1158b22 c4 +1215b23 c4 + 1278b24 c4 + 1335b25 c4 + 1398b26 c4 + 1455b27 c4 + 1518b28 c4 +1575b29 c4 + 1638b30 c4 + 1695b31 c4 + 1758b32 c4 + 1815b33 c4 + 1878b34 c4 +1935b35 c4 + 1998b36 c4 + 2055b37 c4 + 2118b38 c4 + 2175b39 c4 + 2238b40 c4 +2295b41 c4 + 2358b42 c4 + 2415b43 c4 + 2478b44 c4 + 2535b45 c4 + 2598b46 c4 +2655b47 c4 + 2718b48 c4 + 6685b37 c6 + 6878b38 c6 + 7081b39 c6 + 7274b40 c6 +7477b41 c6 + 7670b42 c6 + 7873b43 c6 + 8066b44 c6 + 8269b45 c6 + 8462b46 c6 ) +u10 (5152b28 c6 + 5361b29 c6 u10 + 5580b30 c6 + 5789b31 c6 + 6008b32 c6 +6217b33 c6 + 6436b34 c6 + 6645b35 c6 + 6864b36 c6 + 7073b37 c6 + 7292b38 c6 +7501b39 c6 + 7720b40 c6 + 7929b41 c6 + 8148b42 c6 + 8357b43 c6 +8576b44 c6 ) + u12 (4294b24 c6 + 4522b25 c6 u12 + 4740b26 c6 +4968b27 c6 + 5186b28 c6 + 5414b29 c6 + 5632b30 c6 + 5860b31 c6 +6078b32 c6 + b33 c6 + 6524b34 c6 + 6752b3 5c6 + 6970b36 c6 +7198b37 c6 + 7416b38 c6 + 7644b39 c6 + 7862b40 c6 + 8090b41 c6

100

The Center and Focus Problem +8308b42 c6 + 20412b40 c8 ) + u14 (18369b36 c8 + 18987b37 c8 +19590b38 c8 ) + u16 (15835b32 c8 + 16454b33 c8 + 17088b34 c8

(6.39)

+17707b c + 18341b c ) + · · · 35 8

36 8

There from, an ordinary Hilbert series of the algebra S1 ,2 has the form (the first 61 terms):  (t) = H(S1 ,2 , t, t, t) = 1 + t + 2t2 + 2t3 + 3t4 + 3t5 HS1,2

+4t6 + 4t7 + 5t8 + 5t9 + 6t10 + 6t11 + 7t12 + 7t13 + 76t14 +87t15 + 96t16 + 107t17 + 116t18 + 127t19 + 136t20 + 147t21 +156t22 + 167t23 + 176t24 + 187t25 + 196t26 + 207t27 +1014t28 + 1082t29 + 2142t30 + 2268t31 + 2392t32 + 2518t33 +2642t34 + 2768t35 + 2892t36 + 3018t37 + 3142t38 + 3268t39 40

+3392t +14347t

45

+ 3518t

41

+ 14908t

46

+ 7936t

42

+ 15471t

+ 8290t

47

43

+ 16032t

+ 13784t 48

(6.40)

44

+ 16595t49

+17156t50 + 24404t51 + 25158t52 + 25924t53 + 26678t54 +27444t55 + 44033t56 + 45418t57 + 65175t58 +67178t59 + 89581t60 + · · · We consider the first 61 terms in expansion of Hilbert series of the algebra SI1,2 , which according to [33] are obtained from (4.13)–(4.15) in the following way: HSI1,2 (t) = H(S1,2 , 0, t, t) = 1 + t + 2t2 + 5t3 + 10t4 + 17t5 + 30t6 + 50t7 +81t8 + 125t9 + 188t10 + 276t11 + 399t12 + 559t13 + 772t14 + 1051t15 +1409t16 + 1859t17 + 2428t18 + 3133t19 + 4004t20 + 5064t21 + 6350t22 +7897t23 + 9752t24 + 11947t25 + 14544t26 + 17597t27 + 21168t28 +25315t29 + 30127t30 + 35673t31 + 42051t32 + 49345t33 + 57668t34 +67127t35 + 77855t36 + 89960t37 + 103603t38 + 118928t39 + 136102t40 +155281t41 + 176675t42 + 200462t43 + 226870t44 + 256104t45 +288419t46 + 324057t47 + 363307t48 + 406419t49 + 453726t50 +505532t51 + 562185t52 + 624013t53 + 691426t54 + 764788t55 +844540t56 + 931088t57 + 1024916t58 + 1126484t59 +1236327t60 + · · · Since for series (6.40) and (6.41) there is an inequality HS1 ,2 (t) ≤ HSI1,2 (t),

(6.41)

Algebraically Independent Focus Quantities

101

then in assumption that this inequality holds for the remaining terms of the considered series, we obtain the inequality (S1 ,2 ) ≤ (SI1,2 ). Note that S1 ,2 is not a subalgebra in SI1,2 . Since from [33] we have (SI1,2 ) = 7, then according to the last inequality we obtain Remark 6.1. One of the ways to improve the upper bound of a number of algebraically independent generalized focus pseudo-quantities (as well as focus quantities) for differential system (6.1), that take part in solving the center and focus problem for a given differential system, is in the supposed inequality (S1 ,2 ) ≤ 7. But, on the other hand, you can easily check with (6.40), that for the first 61 terms from (6.40), we have HS1 ,2 (t)
8 in identity (5.12). For obtaining the quantity G1 , we write system (6.45) in the matrix form A1 B1 = C1 , where ⎛ ⎜ ⎜ ⎜ A1 = ⎜ ⎝

4c 4d 0 0 0 ⎛

⎜ ⎜ ⎜ ⎜ B1 = ⎜ ⎜ ⎝

4e 12c + 4f 12d 0 0 ⎞

b0 b1 b2 b3 b4 G1

0 12e 12c + 12f 12d 0

0 0 12e 4c + 12f 4d

(6.50)

0 0 0 4e 4f

−e2 2ce − 2ef −c2 + 2de + 2cf − f 2 −2cd + 2df −d2

⎛ 2ep − ct + f t ⎟ ⎟ ⎜ cp + f p + 6eq − 2dt − 3cu + 3f u ⎟ ⎜ ⎜ ⎟ ⎟ , C1 = ⎜ −3cq + 3f q + 6er − 6du − 3cv + 3f v ⎟ ⎝ −3cr + 3f r + 2es − 6dv − cw + f w ⎠ −cs + f s − 2dw

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

(6.51)

Since the dimension of the matrix A1 is 5 × 6, then it is clear that we have at least one free variable. Therefore, choosing one of bi (i ∈ {0, , 1, ..., 4}) as a free variable and using the Cramer’s rule for system (6.50), for each fixed i, we obtain G1,i + B1,i bi G1 = , (6.52) σ1,i

Algebraically Independent Focus Quantities

105

where G1,i , B1,i , σ1,i (see, Appendix 5) are polynomials in coefficients of system (6.42), and bi are undetermined coefficients of the function U (x, y) from (6.49). In the future, we will need an explicit form of the operators X1 , ..., X4 of the Lie algebra L4 for system (6.42), the expressions of which are obtained from Section 1.5: X1 = x

∂ ∂ ∂ ∂ + D1 , X2 = y + D2 , X3 = x + D3 , X4 = y + D4 , ∂x ∂x ∂y ∂y (6.53)

where ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ −e − 2p −q +s − 3t − 2u −v , ∂d ∂e ∂p ∂q ∂s ∂t ∂u ∂v ∂ ∂ ∂ ∂ ∂ ∂ D2 = e + (f − c) −e +t + (u − p) + (v − 2q) ∂c ∂d ∂f ∂p ∂q ∂r ∂ ∂ ∂ ∂ +(w − 3r) −t − 2u − 3v , ∂s ∂u ∂v ∂w (6.54) ∂ ∂ ∂ ∂ ∂ ∂ ∂ D3 = −d + (c − f ) +d − 3q − 2r −s + (p − 3u) ∂c ∂e ∂f ∂p ∂q ∂r ∂t ∂ ∂ ∂ +(q − 2v) + (r − w) +s , ∂u ∂v ∂w ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ D4 = −d +e −q − 2r − 3s +t −v − 2w . ∂d ∂e ∂q ∂r ∂s ∂t ∂v ∂w D1 = d

By studying matrices (6.51) of system (6.50), we conclude that G1,i from (6.52) are homogeneous polynomials of degree 5 with respect to the linear part, and of degree 1 with respect to the cubic part of system (6.42). Note that G1,i from (6.52) are homogeneous polynomials in coefficients of system (6.42), where for i = 0, 1, 2, 3, 4 they are polynomials of isobarities with weights, respectively (see, Appendix 5): (3, −1), (2, 0), (1, 1), (0, 2), (−1, 3).

(6.55)

According to formula (3.3) (for differential system (6.42) and the theory of invariants of differential systems [37,43]), it follows that the numerators of fractions (6.52) can be coefficients in comitants of the weight −1 of the type (4, 5, 1). This means that according to (2.56) with the help of Lie differential operator D3 from (6.54) for differential system (6.42) and numerator of fraction (6.52), we obtain a system of four linear nonhomogeneous partial differential equations: D3 (G1,0 + B1,0 b0 ) = G1,1 + B1,1 b1 , D3 (G1,1 + B1,1 b1 ) = −G1,2 − B1,2 b2 , −D3 (G1,2 + B1,2 b2 ) = G1,3 + B1,3 b3 , D3 (G1,3 + B1,3 b3 ) = −G1,4 − B1,4 b4 , (6.56)

106

The Center and Focus Problem

with five unknowns b0 , b1 , b2 , b3 , b4 . According to Lemma 2.4, system (6.56) has an infinite number of solutions. Note that a particular solution of this system is b0 = b1 = b2 = b3 = b4 = 0, for which the polynomial f4 (x, y) = G1,0 x4 + 4G1,1 x3 y + 2G1,2 x2 y 2 + 4G1,3 xy 3 + G1,4 y 4

(6.57)

is a centro-affine comitant of differential system (6.42). This fact is also confirmed by Theorem 2.2 with the operators X1 − X4 from (6.54) for differential system (6.42), for which we have the equalities X1 (f4 ) = X4 (f4 ) = f4 , X2 (f4 ) = X3 (f4 ) = 0. It is evident that differential system (6.56) has an infinite number of solutions b0 , b1 , b2 , b3 , b4 , which define centro-affine comitants of type f4 (x, y) = (G1,0 + B1,0 b0 )x4 + 4(G1,1 + B1,1 b1 )x3 y + 2(G1,2 + B1,2 b2 )x2 y 2 + 4(G1,3 + B1,3 b3 )xy 3 + (G1,4 + B1,4 b4 )y 4 .

(6.58)

In view of the abovementioned, comitant (6.57) or (6.58) belongs to the linear space (4,5,1) (6.59) S1,3 , of Sibirsky algebra S1,3 . Note that comitants (6.57) on the invariant variety V from (5.11) for differential system (6.42) have the form f4 (x, y)|V = 6(p + r + u + w)(x2 + y 2 )2 (G1 |V = 6(p + r + u + w)). (6.60) Similarly to the previous case, for determining the quantity G2 , we write systems of equations (6.45), (6.47) in the matrix form (see, Appendix 6): A2 B2 = C2 ,

(6.61)

from where we obtain G2 =

G2,i,j + B2,i,j bi + D2,i,j dj , (i = 0, 4, j = 0, 6). σ2,i,j

(6.62)

By studying matrix equality (6.61), we obtain that degG2,i,j = 14, and using systems (6.45), (6.47), we obtain that G2,i,j from (6.62) has the type (0, 12, 2), i.e. G2,i,j are homogeneous polynomials of degree 12 in coefficients of the linear part and of degree 2 in coefficients of the cubic part of the differential system s(1, 3) from (6.42). Computing the expressions G2,i,j for each i = 0, 4 and j = 0, 6, we obtain for their isobarities the following Table 6.2 (see, Appendix 6). Table 6.2. Isobarities with weight of polynomials G2,i,j for the system s(1, 3) Note that for j = 0, 6 we have

Algebraically Independent Focus Quantities G 2,i,j b0 b1 b2 b3 b4

G2 ≡

d0 (7,−3) (6,−2) (5,−1) (4,0) (3,1)

d1 (6,−2) (5,−1) (4,0) (3,1) (2,2)

d2 (5,−1) (4,0) (3,1) (2,2) (1,3)

d3 d4 (4,0) (3,1) (3,1) (2,2) (2,2) (1,3) (1,3) (0,4) (0,4) (−1,5)

107 d5 d6 (2,2) (1,3) (1,3) (0,4) (0,4) (−1,5) (−1,5) (−2,6) (−2,6) (−3,7)

G2,2,j 3 |V = (−11pq − 15qr + 5ps + rs − pt − 5rt − 3qu σ2,2,j 32

(6.63)

+5su − tu + 7pv + 3rv + 15uv − 7qw + sw − 5tw + 11vw). From the set G2,2,j we choose the expression G2,2,0 as a semi-invariant, which according to Table 6.2 has the weight −1. From here, using (2.56) and (6.62), we obtain that the comitant corresponding to the quantity G2 belongs to the type (6, 12, 2). Similarly to the previous case, we choose a comitant of the weight −1 of the differential system s(1, 3) from (6.42), which contains the expressions G2,2,j + B2,2,j b2 + D2,2,j dj (j = 0, 6) as a semi-invariant, and we find that it belongs to the linear space (6,12,2) (6.64) , S1,3 of Sibirsky algebra S1,3 . Consider determination of G3 of homogeneity of degree 8 with respect to the phase variables x and y in (6.62). Writing the system that consists of (6.45), (6.47), (6.49) in the matrix form A3 B3 = C3 , we obtain G3 =

G3,i,j,k + B3,i,j,k bi + D3,i,j,k dj + F3,i,j,k fk σ3,i,j,k

(6.65)

(i = 0, 4; j = 0, 6; k = 0, 8). Similarly to the previous case, we choose a comitant of the weight −1 of the differential system s(1, 3) from (6.42), which contains the expressions G3,2,j,k + B3,2,j,k b2 + D3,2,j,k dj + F3,2,j,k fk (k = 0, 8) as a semi-invariant, and we find that it belongs to the linear space (8,22,3)

S1,3

(6.66)

of Sibirsky algebra S1,3 . Let us consider the extension of system (6.44)–(6.49), which contains the quantity Gk , and is obtained from (5.12) for differential system (6.42) and function (6.43). We write system (6.44)–(6.49) in the matrix form Ak Bk = Ck . We denote by mGk the number of equations and by nGk the number of

108

The Center and Focus Problem

unknowns of this system. Note that these numbers are written as follows: mGk = '()* 5 +7 +9 +... + 2k + 3, G1

' () * '

G2

()

G3 . . .

'

* ()

*

Gk

nGk = '()* 6 +9 +12 +... + (2k + 3) + k, G1

' () * '

G2

()

*

G3 . . .

'

()

*

Gk

for k = 1, 2, 3, . . .. From here we obtain mGk = k(k + 4),

(6.67)

and nGk = mGk + k. From these systems, it is obtained Gk =

Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik . σk,i1 ,i2 ,...,ik

(6.68)

It is important to determine the degree of the polynomial Gk,i1 ,i2 ,...,ik + +Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik in coefficients of differential system (6.42). Note that the degree of nonzero coefficient of the polynomial Gi (i = 1, k) in coefficients of differential system (6.42) in Cramer’s determinant of order mGk , when the last column corresponding to the quantity Gk is replaced with the column corresponding to free members, forms the following diagram G1 , G2 , G3 , ..., Gk−1 , Gk . ↓

↓

↓

↓

↓

2 3 4 k 2 Then the degree of the polynomials Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik in coefficients of differential system (6.42), denoted by NGk , will be written as k(k − 1) NGk = mGk + + 1, 2 from where we obtain 1 (6.69) NGk = (3k 2 + 7k + 2). 2

Algebraically Independent Focus Quantities

109

It is the degree of homogeneity of polynomials Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik in coefficients of system (6.42), which is contained in a polynomial of type (δ, d0 , d1 ), where δ is the degree of homogeneity of polynomial in x and y, d0 is the degree of homogeneity of polynomial in coefficients of linear part, and d1 is the degree of homogeneity of polynomial in coefficients of the cubic part of the system s(1, 3) from (6.42). Since δ = 2(k + 1) and d1 = k, then d0 = NGk − 2k. Based on this, we find that a comitant of the weight −1 of the differential system s(1, 3) from (6.42), that contains Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik as a semi-invariant and corresponds to the quantity Gk for k = 1, 2, 3, ..., belongs to the type 

1 2(k + 1), (3k 2 + 5k + 2), k , 2

(6.70)

where 2(k + 1) is the degree of homogeneity of the comitant in phase variables 1 x, y; (3k 2 + 5k + 2) is the degree of homogeneity of comitant in the coeffi2 cients c, d, e, f of linear part; k is the degree of homogeneity of the comitant in the coefficients p, q, r, s, t, u, v, w of cubic part of the differential system s(1, 3) from (6.42). Therefore, the expressions Gk,i1 ,i2 ,...,ik +Bk,i1 ,i2 ,...,ik bi1 +· · · +Zk,i1 ,i2 ,...,ik zik , which determine comitants of types (6.70), corresponding to the quantity Gk (k = 1, 2, 3, ...), will be called generalized focus pseudo-quantities, and comitants of type (6.70) for k = 1, 2, 3, ... will be called comitants which contain generalized focus pseudo-quantities Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik as coefficients. 1

2

Note that the spaces S (2(k+1), 2 (3k +5k+2),k) are generalized records of spaces (6.59), (6.64), (6.66) for k = 1, 2, 3, ... of Sibirsky algebra S1,3 .

6.3

On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities, that Take Part in Solving the Center and Focus Problem for the Differential System s(1, 3)

Consider the differential system s(1, 3) from (6.42). In this case, according to the paper [33], using Theorem 4.4, there takes place Theorem 6.3. Dimension of linear space of centro-affine comitants of type (d) = (δ, d0 , d1 ) for the differential system s(1, 3) from (6.42), denoted by

110

The Center and Focus Problem (d)

dimR V1,3 , is equal to the coefficient of the monomial uδ bd0 dd1 in decomposition of generalized Hilbert series from (4.22)–(4.24) for Sibirsky algebra S1,3 of comitants of the considered system. Consider the subalgebra S1 ,3 ⊂ S1,3 , which we write in the form S1 ,3 =

#

(d )

S1,3 ,

(6.71)

(d) (d )

where by S1,3 the following linear spaces are denoted: (0,0,0)

S1,3

(0,1,0)

= R, S1,3

(2(k+1), 12 (3k2 +5k+2),k)

, ..., S1,3

, k = 1, 2, ...,

(6.72)

as well as spaces from S1,3 , which contain all kinds of their products. Since the algebra S1 ,3 is a graded subalgebra in a finitely defined algebra S1,3 , then according to Proposition 4.1, we obtain (S1 ,3 ) ≤ (S1,3 ). From this inequality and from the fact that according to Theorem 4.5, we have (S1,3 ) = 11, according to Remark 2.3 on semi-invariants and the fact that generalized focus pseudo-quantities are coefficients of some comitants, the following is true: Theorem 6.4. Maximal number of algebraically independent generalized focus pseudo-quantities in the center and focus problem for differential system (6.42) does not exceed 11. According to Proposition 4.2, Observation 5.2 and equality (5.14), it follows that the maximal number of algebraically independent focus quantities Lk (k = 1, ∞) cannot exceed maximal number of algebraically independent generalized focus pseudo-quantities Gk,i1 ,i2 ,...,ik + Bk,i1 ,i2 ,...,ik bi1 + · · · + Zk,i1 ,i2 ,...,ik zik . Hence, according to Theorem 6.4, we have Consequence 6.2. Upper bound of the number of algebraically independent focus quantities that take part in solving the center and focus problem for differential system (6.42) does not exceed 11. (d )

Consider types of spaces S1,3 from (6.72) for d = δ  + d0 + d1 ≤ 50, which are obtained from expansion of the following fraction in a power series: 1 (1 − b)(1 − 





u4 b5 d)(1

−

u6 b12 d2 )(1

− u8 b22 d3 )(1 − u10 b35 d4 ) (d )

,

(6.73)

where uδ bd0 dd1 shows the type of space S1,3 for (d ) = (δ  , d0 , d1 ). In consideration of these types and generalized Hilbert series (4.22)–(4.24) of the  algebra S1,3 , we can write expansion of Hilbert series of the algebra S1,3 for d = δ  + d0 + d1 ≤ 50, which has the form

Algebraically Independent Focus Quantities

111

 H(S1,3 , u, b, d) = 1 + b + 2b2 + 2b3 + 3b4 + 3b5 + 4b6 + 4b7 + 5b8 + 5b9

+6b10 + 6b11 + 7b12 + 7b13 + 8b14 + 8b15 + 9b16 + 9b17 + 10b18 + 10b19 +11b20 + 11b21 + 12b22 + 12b23 + 13b24 + 13b25 + 14b26 + 14b27 + 15b28 +15b29 + 16b30 + 16b31 + 17b32 + 17b33 + 18b34 + 18b35 + 19b36 + 19b37 +20b38 + 20b39 + 21b40 + 21b41 + 22b42 + 22b43 + 23b44 + 23b45 + 24b46 +24b47 + 25b48 + 25b49 + 26b50 + (18b5 d + 22b6 d + 26b7 d + 30b8 d + 34b9 d +38b10 d + 42b11 d + 46b12 d + 50b13 d + 54b14 d + 58b15 d + 62b16 d +66b17 d + 70b18 d + 74b19 d + 78b20 d + 82b21 d + 86b22 d + 90b23 d +94b24 d + 98b25 d + 102b26 d + 106b27 d + 110b28 d + 114b29 d +118b30 d + 122b31 d + 126b32 d + 130b33 d + 134b34 d + 138b35 d +142b36 d + 146b37 d + 150b38 d + 154b39 d + 158b40 d + 162b41 d + 166b42 d +170b43 d + 174b44 d + 178b45 d)u4 + (174b12 d2 + 193b13 d2 + 208b14 d2 +227b15 d2 + 242b16 d2 + 261b17 d2 + 276b18 d2 + 295b19 d2 + 310b20 d2 +329b21 d2 + 344b22 d2 + 363b23 d2 + 378b24 d2 + 397b25 d2 + 412b26 d2 +431b27 d2 + 446b28 d2 + 465b29 d2 + 480b30 d2 + 499b31 d2 +514b32 d2 + 533b33 d2 + 548b34 d2 + 567b35 d2 + 582b36 d2 +601b37 d2 + 616b38 d2 + 635b39 d2 + 650b40 d2 + 669b41 d2 +684b42 d2 )u6 + (136b10 d2 + 152b11 d2 + 172b12 d2 + 188b13 d2 +208b14 d2 + 224b15 d2 + 244b16 d2 + 260b17 d2 + 280b18 d2 +296b19 d2 + 316b20 d2 + 332b21 d2 + 352b22 d2 + 368b23 d2 +388b24 d2 + 404b25 d2 + 424b26 d2 + 440b27 d2 + 460b28 d2 +476b29 d2 + 496b30 d2 + 512b31 d2 + 532b32 d2 + 548b33 d2 +568b34 d2 + 584b35 d2 + 604b36 d2 + 620b37 d2 + 640b38 d2 +656b39 d2 + 676b40 d2 + 1098b22 d3 + 1155b23 d3 + 1212b24 d3 +1269b25 d3 + 1326b26 d3 + 1383b27 d3 + 1440b28 d3 + 1497b29 d3 +1554b30 d3 + 1611b31 d3 + 1668b32 d3 + 1725b33 d3 + 1782b34 d3 +1839b35 d3 + 1896b36 d3 + 1953b37 d3 + 2010b38 d3 + 2067b39 d3 )u8 +(791b17 d3 + 850b18 d3 + 909b19 d3 + 968b20 d3 + 1027b21 d3 +1086b22 d3 + 1145b23 d3 + 1204b24 d3 + 1263b25 d3 + 1322b26 d3 +1381b27 d3 + 1440b28 d3 + 1499b29 d3 + 1558b30 d3 + 1617b31 d3 +1676b32 d3 + 1735b33 d3 + 1794b34 d3 + 1853b35 d3 + 1912b36 d3 +1971b37 d3 + 4904b35 d4 + 5056b36 d4 )u10 + (630b15 d3 + 690b16 d3 +750b17 d3 + 810b18 d3 + 870b19 d3 + 930b20 d3 + 990b21 d3 + 1050b22 d3 +1110b23 d3 + 1170b24 d3 + 1230b25 d3 + 1290b26 d3 + 1350b27 d3

112

The Center and Focus Problem +1410b28 d3 + 1470b29 d3 + 1530b30 d3 + 1590b31 d3 + 1650b32 d3 +1710b33 d3 + 1770b34 d3 + 1830b35 d3 + 3142b24 d4 + 3299b25 d4 +3466b26 d4 + 3623b27 d4 + 3790b28 d4 + 3947b29 d4 + 4114b30 d4

+4271b31 d4 + 4438b32 d4 + 4595b33 d4 + 4762b34 d4 )u12 + (2696b22 d4 +2865b23 d4 + 3024b24 d4 + 3193b25 d4 + 3352b26 d4 + 3521b27 d4 28 4

29 4

30 4

31 4

32 4

+3680b d + 3849b d + 4008b d + 4177b d + 4336b d )u

(6.74) 14

+(2230b20 d4 + 2390b21 d4 + 2560b22 d4 + 2720b23 d4 + 2890b24 d4 +3050b25 d4 + 3220b26 d4 + 3380b27 d4 + 3550b28 d4 + 3710b29 d4 +3880b30 d4 + 8817b29 d5 )u16 + 7693b27 d5 u18 + 6534b25 d5 u20 + ... From here, an ordinary Hilbert series of the algebra S1 ,3 has the form (the first 51 terms):   HS1,3 (t) = H(S1,3 , t, t, t) = 1 + t + 2t2 + 2t3 + 3t4 + 3t5 + 4t6

+4t7 + 5t8 + 5t9 + 24t10 + 28t11 + 33t12 + 37t13 + 42t14 + 46t15 +51t16 + 55t17 + 60t18 + 64t19 + 379t20 + 418t21 + 458t22 +497t23 + 537t24 + 576t25 + 616t26 + 655t27 + 695t28 + 734t29 +2195t30 + 2353t31 + 2512t32 + 3768t33 + 3984t34 + 4199t35 +4415t36 + 4630t37 + 4846t38 + 5061t39 + 13345t40 + 14046t41 +14758t42 + 15459t43 + 16171t44 + 16872t45 + 17584t46

(6.75)

+18285t47 + 18997t48 + 24602t49 + 48510t50 + ... We consider the first 51 terms in expansion of Hilbert series of the algebra SI1,3 , which according to [33] are obtained from (4.22)–(4.24) in the following way: HSI1,3 (t) = H(S1,3 , 0, t, t) = 1 + t + 5t2 + 9t3 + 24t4 + 42t5 + 95t6 + 160t7 +308t8 + 506t9 + 877t10 + 1376t11 + 2229t12 + 3358t13 + 5144t14 +7498t15 + 10996t16 + 15545t17 + 22032t18 + 30335t19 + 41764t20 +56226t21 + 75544t22 + 99686t23 + 131205t24 + 170114t25 + 219901t26 +280744t27 + 357236t28 + 449800t29 + 564495t30 + 702002t31 + 870184t32 +1070195t33 + 1311989t34 + 1597351t35 + 1938881t36 + 2339064t37 +2813664t38 + 3366216t39 + 4016096t40 + 4768162t41 + 5646208t42 +6656574t43 + 7828224t44 + 9169512t45 + 10715232t46 + 12476184t47 +14494113t48 + 16782555t49 + 19391253t50 + ... (6.76) Since for series (6.75) and (6.76), the following inequality holds HS1 ,3 (t) ≤ HSI1,3 (t),

Algebraically Independent Focus Quantities

113

then in assumption that this inequality holds for remaining terms of the considered series, we obtain the inequality (S1 ,3 ) ≤ (SI1,3 ). Note that S1 ,3 is not a subalgebra in SI1,3 . Since from Theorem 4.5, we have (SI1,3 ) = 9, then according to the last inequality, we obtain Remark 6.2. One of the ways to improve the upper bound of a number of algebraically independent generalized focus pseudo-quantities (as well as focus quantities) for differential system (6.42), that take part in solving the center and focus problem for a given differential system, is in the supposed inequality  ) ≤ 9. (S1,3 However, on the other hand, you can easily check with (6.75), that for the first 51 terms, we have 1 HS1 ,3 (t) < . (1 − t)7 If we assume that this inequality is true for all terms of series HS1 ,3 (t) and (1 − t)−7 , then perhaps there is an improvement in the majorant assessment of the maximal number of algebraically independent focus quantities that take part in solving the center and focus problem for the differential system s(1, 3) from (6.42), which is expressed by the inequality (S1 ,3 ) < 7.

6.4

The Differential System s(1, 4) and Algebraically Independent Generalized Focus Pseudo-Quantities

Consider the differential system s(1, 4), which we write in the form x˙ = cx + dy + gx4 + 4hx3 y + 6ix2 y 2 + 4jxy 3 + ky 4 , y˙ = ex + f y + lx4 + 4mx3 y + 6nx2 y 2 + 4oxy 3 + py 4

(6.77)

with a infinitely defined graded algebra of unimodular comitants S1,4 [33]. For this system, we write function (5.13) in the form U = k2 + a0 x3 + 3a1 x2 y + 3a2 xy 2 + a3 y 3 + b0 x4 + 4b1 x3 y + 6b2 x2 y 2 +4b3 xy 3 + b4 y 4 + c0 x5 + 5c1 x4 y + 10c2 x3 y 2 + 10c3 x2 y 3 + 5c4 xy 4 +c5 y 5 + d0 x6 + 6d1 x5 y + 15d2 x4 y 2 + 20d3 x3 y 3 + 15d4 x2 y 4 + 6d5 xy 5 +d6 y 6 + e0 x7 + 7e1 x6 y + 21e2 x5 y 2 + 35e3 x4 y 3 + 21e5 x2 y 5 + 7e6 xy 6 +e7 y 7 + f0 x8 + 8f1 x7 y + 28f2 x6 y 2 + 56f3 x5 y 3 + 70f4 y 4 + 56f5 x3 y 5 +28f6 x2 y 6 + 8f7 xy 7 + f8 y 8 + g0 x9 + 9g1 x8 y + 36g2 x7 y 2 + 84g3 x6 y 3

114

The Center and Focus Problem +126g4 x5 y 4 + 126g5 x4 y 5 + 84g6 x3 y 6 + 36g7 x2 y 7 + 9g8 xy 8 + g9 y 9 +h0 x10 + 10h1 x9 y + 45h2 x8 y 2 + 120h3 x7 y 3 + 210h4 x6 y 4 + 252h5 x5 y 5 +210h6 x4 y 6 + 120h7 x3 y 7 + 45h8 x2 y 8 + 10h9 xy 9 + h10 y 10 + i0 x11

+11i1 x10 y + 55i2 x9 y 2 + 165i3 x8 y 3 + 330i4 x7 y 4 + 462i5 x6 y 5 + 462i6 x5 y 6 +330i7 x4 y 7 + 55i9 x2 y 9 + 11i10 xy 10 + i11 y 11 + j0 x12 + 12j1 x11 y +165i8 x3 y 8 + 66j2 x10 y 2 + 220j3 x9 y 3 + 495j4 x8 y 4 + 792j5 x7 y 5 +924j6 x6 y 6 + 792j7 x5 y 7 + 495j8 x4 y 8 + 220j9 x3 y 9 + 66j10 x2 y 10 +12j11 xy 11 + j12 y 12 + k0 x13 + 13k1 x12 y + 78k2 x11 y 2 + 286k3 x10 y 3 +715k4 x9 y 4 + 1287k5 x8 y 5 + 1716k6 x7 y 6 + 1716k7 x6 y 7 + 1287k8 x5 y 8 +715k9 x4 y 9 + 286k10 x3 y 10 + 78k11 x2 y 11 + 13k12 xy 12 + k13 y 13 +l0 x14 + 14l1 x13 y + 91l2 x12 y 2 + 364l3 x11 y 3 + 1001l4 x10 y 4 +2002l5 x9 y 5 + 3003l6 x8 y 6 + 3432l7 x7 y 7 + 3003l8 x6 y 8 5 9

4 10

+2002l9 x y + 1001l10 x y +14l13 xy

13

3 11

+ 364l11 x y

+ l14 y

14

(6.78)

2 12

+ 91l12 x y

+ ...,

where k2 ≡ 0 is from (5.8), and a0 , a1 , ..., l13 , l14 , ... are unknown coefficients. Identity (5.12) along the trajectories of differential system (6.77) with function (6.78) splits into the following systems of equations (equality (5.15) is omitted): x3 : 3a0 c + 3a1 e = 0, x2 y : 6a1 c + 3a0 d + 6a2 e + 3a1 f = 0, xy 2 : 3a2 c + 6a1 d + 3a3 e + 6a2 f = 0,

(6.79)

y 3 : 3a2 d + 3a3 f = 0; x4 : 4b0 c + 4b1 e − e2 G1 = 0, x3 y : 12b1 c + 4b0 d + 12b2 e + 4b1 f + 2ceG1 − 2ef G1 = 0, x2 y 2 : 12b2 c + 12b1 d + 12b3 e + 12b2 f − c2 G1 + 2deG1 + 2cf G1 − f 2 G1 = 0, xy 3 : 4b3 c + 12b2 d + 4b4 e + 12b3 f − 2cdG1 + 2df G1 = 0, y 4 : 4b3 d + 4b4 f − d2 G1 = 0; x : 5cc0 + 5c1 e − 2eg + cl − f l = 0, 5

x4 y : 20cc1 + 5c0 d + 20c2 e + 5c1 f + cg − f g − 8eh + 2dl + 4cm − 4f m = 0, x3 y 2 : 30cc2 + 20c1 d + 30c3 e + 20c2 f + 4ch − 4f h − 12ei + 8dm + 6cn − 6f n = 0,

(6.80)

Algebraically Independent Focus Quantities

115

x2 y 3 : 20cc3 + 30c2 d + 20c4 e + 30c3 f + 6ci − 6f i − 8ej + 12dn + 4co − 4f o = 0, xy 4 : 5cc4 + 20c3 d + 5c5 e + 20c4 f + 4cj − 4f j − 2ek + 8do + cp − f p = 0,

(6.81)

y 5 : 5c4 d + 5c5 f + ck − f k + 2dp = 0; x6 : 6cd0 + 6d1 e + 3a0 g + 3a1 l = −e3 G2 , x5 y : 6dd0 + 30cd1 + 30d2 e + 6d1 f + 6a1 g + 12a0 h + 6a2 l + 12a1 m = 3ce2 G2 − 3e2 f G2 , x4 y 2 : 30dd1 + 60cd2 + 60d3 e + 30d2 f + 3a2 g + 24a1 h + 18a0 i + 3a3 l + 24a2 m + 18a1 n = −3c2 eG2 + 3de2 G2 + 6cef G2 − 3ef 2 G2 , x3 y 3 : 60dd2 + 60cd3 + 60d4 e + 60d3 f + 12a2 h + 36a1 i + 12a0 j + 12a3 m + 36a2 n + 12a1 o = c3 G2 − 6cdeG2 − 3c2 f G2

(6.82)

+ 6def G2 + 3cf G2 − f G2 , 2

3

2 4

x y : 60dd3 + 30cd4 + 30d5 e + 60d4 f + 18a2 i + 24a1 j + 3a0 k + 18a3 n + 24a2 o + 3a1 p = 3c2 dG2 − 3d2 eG2 − 6cdf G2 + 3df 2 G2 , xy 5 : 30dd4 + 6cd5 + 6d6 e + 30d5 f + 12a2 j + 6a1 k + 12a3 o + 6a2 p = 3cd2 G2 − 3d2 f G2 , y 6 : 6dd5 + 6d6 f + 3a2 k + 3a3 p = d3 G2 ; x7 : 7ce0 + 7ee1 + 4b0 g + 4b1 l = 0, x6 y : 7de0 + 42ce1 + 42ee2 + 7e1 f + 12b1 g + 16b0 h + 12b2 l + 16b1 m = 0, x5 y 2 : 42de1 + 105ce2 + 105ee3 + 42e2 f + 12b2 g + 48b1 h + 24b0 i + 12b3 l + 48b2 m + 24b1 n = 0, x4 y 3 : 105de2 + 140ce3 + 140ee4 + 105e3 f + 4b3 g + 48b2 h + 72b1 i + 16b0 j + 4b4 l + 48b3 m + 72b2 n + 16b1 o = 0, x3 y 4 : 140de3 + 105ce4 + 105ee5 + 140e4 f + 16b3 h + 72b2 i + 48b1 j + 4b0 k + 16b4 m + 72b3 n + 48b2 o + 4b1 p = 0, x2 y 5 : 105de4 + 42ce5 + 42ee6 + 105e5 f + 24b3 i + 48b2 j + 12b1 k + 24b4 n + 48b3 o + 12b2 p = 0, xy 6 : 42de5 + 7ce6 + 7ee7 + 42e6 f + 16b3 j + 12b2 k + 16b4 o + 12b3 p = 0, y 7 : 7de6 + 7e7 f + 4b3 k + 4b4 p = 0;

(6.83)

116

The Center and Focus Problem x8 : 8cf0 + 8ef1 + 5c0 g + 5c1 l = e4 G3 , x7 y : 8df0 + 56cf1 + 8f f1 + 56ef2 + 20c1 g + 20c0 h + 20c2 l + 20c1 m = −4ce3 G3 + 4e3 f G3 , x6 y 2 : 56df1 + 168cf2 + 56f f2 + 168ef3 + 30c2 g + 80c1 h + 30c0 i + 30c3 l + 80c2 m + 30c1 n = 6c2 e2 G3 − 4de3 G3 − 12ce2 f G3 + 6e2 f 2 G3 , x5 y 3 : 168df2 + 280cf3 + 168f f3 + 280ef4 + 20c3 g + 120c2 h + 120c1 i + 20c0 j + 20c4 l + 120c3 m + 120c2 n + 20c1 o = = −4c3 eG3 + 12cde2 G3 + 12c2 ef G3 − 12de2 f G3 − 12cef 2 G3 + 4ef 3 G3 , x4 y 4 : 280df3 + 280cf4 + 280f f4 + 280ef5 + 5c4 g + 80c3 h + 180c2 i + 80c1 j + 5c0 k + 5c5 l + 80c4 m + 180c3 n + 80c2 o + 5c1 p = c4 G3 − 12c2 deG3 + 6d2 e2 G3 − 4c3 f G3 + 24cdef G3 + 6c2 f 2 G3 − 12def 2 G3 − 4cf 3 G3 + f 4 G3 , x3 y 5 : 280df4 + 168cf5 + 280f f5 + 168ef6 + 20c4 h + 120c3 i + 120c2 j + 20c1 k + 20c5 m + 120c4 n + 120c3 o + 20c2 p = = 4c3 dG3 − 12cd2 eG3 − 12c2 df G3 + 12d2 ef G3 + 12cdf 2 G3 − 4df 3 G3 , x2 y 6 : 168df5 + 56cf6 + 168f f6 + 56ef7 + 30c4 i + 80c3 j + 30c2 k + 30c5 n + 80c4 o + 30c3 p = 6c2 d2 G3 − 4d3 eG3 − 12cd2 f G3 + 6d2 f 2 G3 , xy 7 : 56df6 + 8cf7 + 56f f7 + 8ef8 + 20c4 j + 20c3 k + 20c5 o + 20c4 p = 4cd3 G3 − 4d3 f G3 , y 8 : 8df7 + 8f f8 + 5c4 k + 5c5 p = d4 G3 ; x9 : 6d0 g + 9cg0 + 9eg1 + 6d1 l = 0, x8 y : 30d1 g + 9dg0 + 72cg1 + 9f g1 + 72eg2 + 24d0 h + 30d2 l + 24d1 m = 0, x7 y 2 : 60d2 g + 72dg1 + 252cg2 + 72f g2 + 252eg3 + 120d1 h + 36d0 i + 60d3 l + 120d2 m + 36d1 n = 0, x6 y 3 : 60d3 g + 252dg2 + 504cg3 + 252f g3 + 504eg4 + 240d2 h + 180d1 i + 24d0 j + 60d4 l + 240d3 m + 180d2 n + 24d1 o = 0, x5 y 4 : 30d4 g + 504dg3 + 630cg4 + 504f g4 + 630eg5 + 240d3 h + 360d2 i + 120d1 j + 6d0 k + 30d5 l + 240d4 m + 360d3 n + 120d2 o + 6d1 p = 0,

(6.84)

Algebraically Independent Focus Quantities

117

x4 y 5 : 6d5 g + 630dg4 + 504cg5 + 630f g5 + 504eg6 + 120d4 h + 360d3 i + 240d2 j + 30d1 k + 6d6 l + 120d5 m + 360d4 n + 240d3 o + 30d2 p = 0, x3 y 6 : 504dg5 + 252cg6 + 504f g6 + 252eg7 + 24d5 h + 180d4 i + 240d3 j + 60d2 k + 24d6 m + 180d5 n + 240d4 o + 60d3 p = 0, x2 y 7 : 252dg6 + 72cg7 + 252f g7 + 72eg8 + 36d5 i + 120d4 j + 60d3 k + 36d6 n + 120d5 o + 60d4 p = 0,

(6.85)

xy 8 : 72dg7 + 9cg8 + 72f g8 + 9eg9 + 24d5 j + 30d4 k + 24d6 o + 30d5 p = 0, y 9 : 9dg8 + 9f g9 + 6d5 k + 6d6 p = 0; x10 : 7e0 g + e5 G4 + 10ch0 + 10eh1 + 7e1 l = 0, x9 y : 42e1 g − 5ce4 G4 + 5e4 f G4 + 28e0 h + 10dh0 + 90ch1 + 10f h1 + 90eh2 + 42e2 l + 28e1 m = 0, x8 y 2 : 105e2 g + 10c2 e3 G4 − 5de4 G4 − 20ce3 f G4 + 10e3 f 2 G4 + 168e1 h + 90dh1 + 360ch2 + 90f h2 + 360eh3 + 42e0 i + 105e3 l + 168e2 m 42e1 n = 0, x7 y 3 : 140e3 g − 10c3 e2 G4 + 20cde3 G4 + 30c2 e2 f G4 − 20de3 f G4 + 30ce2 f 2 G4 + 10e2 f 3 G4 + 420e2 h + 360dh2 + 840ch3 + 360f h3 + 840eh4 + 252e1 i + 28e0 j + 140e4 l + 420e3 m + 252e2 n + 28e1 o = 0, x6 y 4 : 105e4 g + 5c4 eG4 − 30c2 de2 G4 + 10d2 e3 G4 − 20c3 ef G4 + 60cde2 f G4 + 30c2 ef 2 G4 − 30de2 f 2 G4 − 20cef 3 G4 + 5ef 4 G4 + 560e3 h + 840dh3 + 1260ch4 + 840f h4 + 1260eh5 + 630e2 i + 168e1 j + 7e0 k + 105e5 l + 560e4 m + 630e3 n + 168e2 o + 7e1 p = 0, x5 y 5 : 42e5 g − c5 G4 + 20c3 deG4 − 30cd2 e2 G4 + 5c4 f G4 − 60c2 def G4 + 30d2 e2 f G4 − 10c3 f 2 G4 + 60cdef 2 G4 + 10c2 f 3 G4 − 20def 3 G4 − 5cf 4 G4 + f 5 G4 + 420e4 h + 1260dh4 + 1260ch5 + 1260f h5 + 1260eh6 + 840e3 i + 420e2 j + 42e1 k + 42e6 l + 420e5 m + 840e4 n + 420e3 o + 42e2 p = 0, x4 y 6 : 7e6 g − 5c4 dG4 + 30c2 d2 eG4 − 10d3 e2 G4 + 20c3 df G4 − 60cd2 ef G4 − 30c2 df 2 G4 + 30d2 ef 2 G4 + 20cdf 3 G4 − 5df 4 G4 + 168e5 h + 1260dh5 + 840ch6 + 1260f h6 + 840eh7 + 630e4 i + 560e3 j + 105e2 k + 7e7 l + 168e6 m + 630e5 n + 560e4 o + 105e3 p = 0,

118

The Center and Focus Problem x3 y 7 : 20cd3 eG4 − 10c3 d2 G4 + 30c2 d2 f G4 − 20d3 ef G4 − 30cd2 f 2 G4 + 10d2 f 3 G4 + 28e6 h + 840dh6 + 360ch7 + 840f h7 + 360eh8 + 252e5 i + 420e4 j + 140e3 k + 28e7 m + 252e6 n + 420e5 o + 140e4 p = 0, x2 y 8 : 5d4 eG4 − 10c2 d3 G4 + 20cd3 f G4 − 10d3 f 2 G4 + 360dh7 + +90ch8 + 360f h8 + 90eh9 + 42e6 i + 168e5 j + 105e4 k + 42e7 n + 168e6 o + 105e5 p = 0,

(6.86)

xy 9 : 5d4 f G4 − 5cd4 G4 + 10eh10 + 90dh8 + 10ch9 + 90f h9 + 28e6 j + 42e5 k + 28e7 o + 42e6 p = 0, y 10 : 10f h10 − d5 G4 + 10dh9 + 7e6 k + 7e7 p = 0; It is evident that linear systems of equations (6.79)–(6.86) in the variables a0 , a1 , a2 , a3 , b0 , b1 ,...,b4 , c0 , c1 ,...,c5 , d0 , d1 ,...,d6 , e0 , e1 ,...,e7 , l0 , l1 ,...,l14 , f0 , f1 ,...,f8 ,..., G1 , G2 , G3 , ... can be considered as a single system that can be extended by adding, after the last equation from (6.86), an infinite number of equations, obtained from the equality of coefficients xα y β for α + β > 10 in identity (5.12). For obtaining the quantity G1 consider system (6.80), that contains constants b0 , b1 , b2 , b3 , b4 , G1 . We write the system in the matrix form A1 B 1 = C1 , where

⎛

⎜ ⎜ ⎜ A1 = ⎜ ⎝

4c 4d 0 0 0

4e 12c + 4f 12d 0 0

0 12e 12c + 12f 12d 0 ⎛ b0 ⎜ b1 ⎜ ⎜ b2 ⎜ B1 = ⎜ ⎜ b3 ⎝ b4 G1

(6.87)

⎞ 0 0 −e2 ⎟ 0 0 2ce − 2ef ⎟ 2 2 ⎟ 12e 0 2de + 2cf − c − f ⎟ , ⎠ 4c + 12f 4e −2cd + 2df 2 4d 4f −d ⎞ ⎛ ⎞ 0 ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ 0 ⎟ ⎟ , C1 = ⎜ 0 ⎟ . (6.88) ⎟ ⎜ ⎟ ⎟ ⎝ 0 ⎠ ⎠ 0

Note that the elements of the first five columns of the matrix A1 from (6.88) are linear functions in coefficients of the system s(1, 4) from (6.77), and nonzero elements of the sixth column (in the product (6.87) they correspond to the quantity G1 ) have the degree 2 with respect to these coefficients. Since system (6.80) consists of five equations with six unknowns, one of them can be declared free. As a free variable, in system (6.80), there can be taken one of the unknowns bi for the fixed i = 0, 4. Then, from the system of equations (6.80), we obtain B1,i bi G1 = (6.89) σ1,i

Algebraically Independent Focus Quantities

119

for any fixed i, equal to 0, 1, 2, 3, 4, where B1,i , σ1,i are polynomials in coefficients of system (6.77), and bi are the undetermined coefficients of the function U (x, y) from (6.78). We obtain that b0 = b1 = b2 = b3 = b4 = 0 can be taken as one particular solution for system (6.80). Therefore, G1 can be considered equal to zero. For obtaining the quantity G2 consider systems (6.79), (6.82). The resulting system consists of four equations (6.79) for determining ai , i = 0, 4, to which seven more equations containing G2 are added. Writing these equations in matrix form and performing similar reasoning, as in the abovementioned case, we obtain G2 =

D2,i di , σ2,i

(6.90)

for each i = 0, 6. Since system (6.79) has a solution a0 = a1 = a2 = = a3 = 0, then we obtain that a particular solution of systems (6.79), (6.82) is a0 = a1 = a2 = a3 = d0 = d1 = d2 = d3 = d4 = d5 = d6 = 0. Thus, G2 can be considered equal to zero. For obtaining the quantity G3 , use the matrix equation (see, Appendix 7) A3 B 3 = C3 ,

(6.91)

Since the dimension of the matrix A3 is 15 × 16, then it is clear that we have at least one free variable. Therefore, choosing one of fi (i ∈ {0, 1, ..., 8}) as a free variable and using the Cramer’s rule for system (6.91), for each fixed i, we obtain G3 =

G3,i + F3,i fi , σ3,i

(6.92)

where G3,i , F3,i , σ3,i are polynomials in coefficients of system (6.77), and fi are undetermined coefficients of the function U (x, y) from (6.78). We are interested in the degree of polynomials G3,i with respect to the coefficients of the system s(1, 4) from (6.77). The indicated degree coincides with the degree of the Cramer determinant ΔG3 . So, the degree of G3,i with respect to the coefficients of the system s(1, 4) from (6.77) will be degG3 = 16 for all i = 0, 8. Taking into account the dimension of system (6.91), we obtain that G3,i has type (0, 14, 2), i.e. G3,i is a homogeneous polynomial of degree 14 with respect to the coefficients of the linear part and homogeneous polynomial of degree 2 with respect to the coefficients of the nonhomogeneity of the fourth order of the system s(1, 4) from (6.77). Zero in (0, 14, 2) shows that the expressions G3,i do not contain phase variables x, y. Since G3,i from (6.92) are homogeneous polynomials in coefficients of the differential system s(1, 4) from (6.77), then according to the paper [43], for i = 0, 8, they are polynomials of isobarities with weights, respectively: (7, −1), (6, 0), (5, 1), (4, 2), (3, 3), (2, 4), (1, 5), (0, 6), (−1, 7).

120

The Center and Focus Problem

We also note that σ3,i are polynomials only in coefficients of the linear part c, d, e, f of the system s(1, 4) from (6.77). For differential system (6.77), using the formula of comitant’s weight (3.3), we obtain that numerators of fractions (6.92) can be coefficients in comitants of weight −1 of type (8, 14, 2). Using Lie differential operator D3 from (6.96) for differential system (6.77), we obtain a system of eight linear nonhomogeneous partial differential equations: D3 (G3,0 + F3,0 f0 ) = G3,1 + F3,1 f1 , D3 (G3,1 + F3,1 f1 ) = −G3,2 − F3,2 f2 , −D3 (G3,2 + F3,2 f2 ) = G3,3 + F3,3 f3 , D3 (G3,3 + F3,3 f3 ) = −G3,4 − F3,4 f4 , −D3 (G3,4 + F3,4 f4 ) = G3,5 + F3,5 f5 , D3 (G3,5 + F3,5 f5 ) = −G3,6 − F3,6 f6 , −D3 (G3,6 + F3,6 f6 ) = G3,7 + F3,7 f7 , D3 (G3,7 + F3,7 f7 ) = −G3,8 − F3,8 f8 (6.93) with nine unknowns f0 , f1 , ..., f8 . According to Lemma 2.4, system (6.93) has an infinite number of solutions. Note that a particular solution for this system is f0 = f1 = ... = f8 = 0, for which the polynomial f8 (x, y) = G3,0 x8 − 8G3,1 x7 y − 4G3,2 x6 y 2 + 8G3,3 x5 y 3 +2G3,4 x4 y 4 − 8G3,5 x3 y 5 − 4G3,6 x2 y 6 + 8G3,7 xy 7 + G3,8 y 8

(6.94)

is a centro-affine comitant of differential system (6.77). This fact is also confirmed by Theorem 2.2 with the operators from Section 1.5 of the form ∂ ∂ + D1 , X2 = y + D2 , ∂x ∂x ∂ ∂ + D 3 , X4 = y + D4 , X3 = x ∂y ∂y

X1 = x

(6.95)

where ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ −e − 3g − 2h −i +k − 4l − 3m ∂d ∂e ∂g ∂h ∂i ∂k ∂l ∂m ∂ ∂ −2n −o , ∂n ∂o ∂ ∂ ∂ ∂ ∂ ∂ D2 = e + (f − c) −e +l + (m − g) + (n − 2h) ∂c ∂d ∂f ∂g ∂h ∂i ∂ ∂ ∂ ∂ ∂ ∂ +(o − 3i) + (p − 4j) −l − 2m − 3n − 4o , ∂j ∂k ∂m ∂n ∂o ∂p ∂ ∂ ∂ ∂ ∂ ∂ ∂ +d − 4h − 3i − 2j − k D3 = −d + (c − f ) ∂c ∂e ∂f ∂g ∂h ∂i ∂j ∂ ∂ ∂ ∂ ∂ +(g − 4m) + (h − 3n) + (i − 2o) + (j − p) +k , ∂l ∂m ∂n ∂o ∂p

D1 = d

D4 = −d

∂ ∂ ∂ ∂ ∂ ∂ ∂ +e −h − 2i − 3j − 4k +l ∂d ∂e ∂h ∂i ∂j ∂k ∂l ∂ ∂ ∂ −n − 2o − 3p , ∂n ∂o ∂p

(6.96)

Algebraically Independent Focus Quantities

121

for differential system (6.77), for which the following equalities hold: X1 (f8 ) = X4 (f8 ) = f8 , X2 (f8 ) = X3 (f8 ) = 0. Differential system (6.93) has an infinite number of solutions f0 , f1 , ..., f8 , which define centro-affine comitants of the type (8, 14, 2), which are written as f8 (x, y) = (G3,0 + F3,0 f0 )x8 − 8(G3,1 + F3,1 f1 )x7 y − 4(G3,2 +F3,2 f2 )x6 y 2 + 8(G3,3 + F3,3 f3 )x5 y 3 + 2(G3,4 + F3,4 f4 )x4 y 4 −8(G3,5 + F3,5 f5 )x3 y 5 − 4(G3,6 + F3,6 f6 )x2 y 6 + 8(G3,7

(6.97)

+F3,7 f7 )xy 7 + (G3,8 + F3,8 f8 )y 8 . According to the abovementioned, comitant (6.97) belongs to the linear space (8,14,2) . S1,4 Note that comitant (6.94) on the variety V from (5.11) for the differential system s(1, 4) from (6.77) has the form f8 (x, y)|V = L3 (x2 + y 2 )4 (G3 |V = L3 ),

(6.98)

where L3 = 648(7gh + 18hi + 3gj + 18ij + 3hk + 7jk − 7gl − 3il − 8hm − 7lm −3gn + 3kn − 18mn + 8jo − 3lo − 18no + 3ip + 7kp − 3mp − 7op) is the first nonzero Lyapunov quantity of differential system (6.77) on an invariant variety V . Consider the extension of system (6.79)–(6.86), which contains the quantity G3k , which is obtained from identity (5.12) for differential system (6.77) and function (6.78). System (6.79)–(6.86) is written in matrix form A3k B3k = C3k . We denote by mG3k the number of equations, and by nG3k the number of unknowns of this system. Note that this number is written as mG3k = 6 + 9 + 12 + 15 + 18 + 21 + · · · + 6k + 3(2k + 1), ' () * ' () * ' () * ' () * G3

G6

G9

G3k

for k = 1, 2, 3, . . .. From here, we obtain mG3k = 6k 2 + 9k.

(6.99)

From these systems, we obtain G3 k =

G3k,i1 ,i2 ,...,ik + B3k,i1 ,i2 ,...,ik bi1 + · · · + Z3k,i1 ,i2 ,...,ik zik . σ3k,i1 ,i2 ,...,ik

(6.100)

It is important to determine the degree of the polynomial G3k,i1 ,i2 ,...,ik + + B3k,i1 ,i2 ,...,ik bi1 + · · · + Z3k,i1 ,i2 ,...,ik zik in coefficients of differential system (6.77).

122

The Center and Focus Problem

Note that the degree of nonzero polynomial coefficient of G3i (i = 1, k) in coefficients of differential system (6.77) in Cramer’s determinant of the order mG3k , when the coefficients in G3k are replaced with free members of the considered system, forms the following diagram: G3 , G6 , G9 , ..., G3(k−1) , G3k . ↓

↓

↓

4

7

10

↓

↓

3(k − 1) + 1 2

Then the degree of the polynomials G3k,i1 ,i2 ,...,ik + B3k,i1 ,i2 ,...,ik bi1 + · · · +Z3k,i1 ,i2 ,...,ik zik with respect to the coefficients of differential system (6.77), denoted by NG3k , will be written as   1 2 NG3k = mG3k − k + (3k − 3k + 2) + k , 2 from where we obtain NG3k =

1 (15k 2 + 15k + 2). 2

(6.101)

It is the degree of homogeneity of polynomials G3k,i1 ,i2 ,...,ik + B3k,i1 ,i2 ,...,ik bi1 + · · · + Z3k,i1 ,i2 ,...,ik zik with respect to the coefficients of system (6.77), which are polynomials of type (0, d0 , d1 ), (6.102) where d0 is the degree of homogeneity of the polynomial with respect to the coefficients of the linear part, d1 is the degree of homogeneity of the polynomial with respect to the coefficients of the nonhomogeneity of the fourth-order of the system s(1, 4) from (6.77). Since d1 = 2k, then d0 = NG3k − 2k. Therefore, according to formula (3.3), we obtain that δ = 2(3k + 1), when the weight g = −1. Based on this, we find that a comitant of the weight −1 of the system s(1, 4) from (6.77), which contains G3k,i1 ,i2 ,...,ik + B3k,i1 ,i2 ,...,ik bi1 + · · · + Z3k,i1 ,i2 ,...,ik zik corresponding to the quantity G3k for k = 1, 2, 3, ... (Gn = 0 if n = 3k) as a semi-invariant (coefficient at the highest degree of x), belongs to the type  1 2 2(3k + 1), (15k + 11k + 2), 2k , (6.103) 2 where 2(3k+1) is the degree of homogeneity of the comitant in phase variables 1 x, y; (15k 2 + 11k + 2) is the degree of homogeneity of the comitant in the 2 coefficients of linear part; 2k is the degree of homogeneity of the comitant in the coefficients of the nonhomogeneity of the fourth order of the system s(1, 4) from (6.77).

Algebraically Independent Focus Quantities

123

Consider the differential system s(1, 4) from (6.77). In this case, according to the paper [33] and using Theorem 4.7, the following is true Theorem 6.5. Dimension of linear space of centro-affine comitants of type (d) = (δ, d0 , d1 ) for the differential system s(1, 4) from (6.77), denoted by (d) dimR V1,4 , is equal to the coefficient of the monomial uδ bd0 ed1 in decomposition of generalized Hilbert series from (4.31)–(4.32) for Sibirsky algebra S1,4 of comitants of the considered system. Consider the subalgebra S1 ,4 ⊂ S1,4 , which we write in the form S1 ,4 =

#

(d )

S1,4 ,

(6.104)

(d) (d )

where by S1,4 the following linear spaces are denoted: (0,0,0)

S1,4

(0,1,0)

= R, S1,4

2(3k+1), 12 (15k2 +11k+2),2k

, ..., S1,4

, k = 1, 2, ...,

(6.105)

as well as spaces from S1,4 , which contain all kinds of their products. Since the algebra S1 ,4 is a graded subalgebra in a finitely defined algebra S1,4 , then according to Proposition 4.1, we obtain (S1 ,4 ) ≤ (S1,4 ). From this inequality and Theorem 4.9, we have (S1,4 ) = 13. According to Remark 5.1 on semi-invariants and the fact that generalized focus pseudo-quantities are coefficients of some comitants, there takes place Theorem 6.6. Maximal number of algebraically independent generalized focus pseudo-quantities in the center and focus problem for differential system (6.77) does not exceed 13. From Proposition 4.2 and Remark 5.1, it follows that the maximal number of algebraically independent focus quantities Lk (k = 1, ∞) cannot exceed the maximal number of algebraically independent generalized focus pseudoquantities G3k,i1 ,i2 ,...,ik + B3k,i1 ,i2 ,...,ik bi1 + · · · + Z3k,i1 ,i2 ,...,ik zik . There from, using Theorem 6.6, we obtain Consequence 6.3. Maximal number of algebraically independent focus quantities that take part in solving the center and focus problem for system (6.77) does not exceed 13. The generalized generating function of the space V1,4 , consisting of direct sum of spaces (6.104), can be written as (0,0,0)

Φ(V1,4 , u, b, e) = dimR S1,4 +

∞  k=1

(0,1,0)

+ dimR S1,4

(2(k+1), 12 (15k2 +11k+2),2k) 2(k+1)

dimR S1,4

u

b 1

b 2 (15k

2

+11k+2) 2k

e .

(6.106)

124

The Center and Focus Problem

Using computer, expand the Hilbert series H(S1,4 , u, b, e) in a power series. Then for (6.106), we have Φ(V1,4 , u, b, e) = 1 + b + 153u4 b14 e2 + 4589u6 b42 e4 + 49632u8 b85 e6 + ... 1

+C2(k+1), 12 (15k2 +11k+2),2k u2(k+1) b 2 (15k

2

+11k+2) 2k

e

+ ...,

(6.107) where C2(k+1), 12 (15k2 +11k+2),2k is an undetermined coefficient. Using this generalized generating function and the Hilbert series H(S1,4 , u, b, e) from Theorem 4.7, the first terms up to u8 b85 e6 in the Hilbert series H(S1 ,4 , u, b, e) were obtained: H(S1 ,4 , u, b, e) = 1 + b + 2b2 + 2b3 + 3b4 + 3b5 + 4b6 + 4b7 + 5b8 + 5b9 +6b10 + 6b11 + 7b12 + 7b13 + 8b14 + 8b15 + 9b16 + 9b17 + 10b18 + 10b19 +11b20 + 11b21 + 12b22 + 12b23 + 13b24 + 13b25 + 14b26 + 14b27 + 15b28 +15b29 + 16b30 + 16b31 + 17b32 + 17b33 + 18b34 + 18b35 + 19b36 + 19b37 +20b38 + 20b39 + 21b40 + 21b41 + 22b42 + 22b43 + 23b44 + 23b45 + 24b46 +24b47 + 25b48 + 25b49 + 26b50 + 26b51 + 27b52 + 27b53 + 28b54 + 28b55 +29b56 + 29b57 + 30b58 + 30b59 + 31b60 + 31b61 + 32b62 + 32b63 + 33b64 +33b65 + 34b66 + 34b67 + 35b68 + 35b69 + 36b70 + 36b71 + 37b72 + 37b73 +38b74 + 38b75 + 39b76 + 39b77 + 40b78 + 40b79 + 41b80 + 41b81 + 42b82 +42b83 + 43b84 + 43b85 + 242b14 e2 u4 + 264b15 e2 u4 + 281b16 e2 u4 +303b17 e2 u4 + 320b18 e2 u4 + 342b19 e2 u4 + 359b20 e2 u4 + 381b21 e2 u4 +398b22 e2 u4 + 420b23 e2 u4 + 437b24 e2 u4 + 459b25 e2 u4 + 476b26 e2 u4 +498b27 e2 u4 + 515b28 e2 u4 + 537b29 e2 u4 + 554b30 e2 u4 + 576b31 e2 u4 +593b32 e2 u4 + 615b33 e2 u4 + 632b34 e2 u4 + 654b35 e2 u4 + 671b36 e2 u4 +693b37 e2 u4 + 710b38 e2 u4 + 732b39 e2 u4 + 749b40 e2 u4 + 771b41 e2 u4 +788b42 e2 u4 + 810b43 e2 u4 + 827b44 e2 u4 + 849b45 e2 u4 + 866b46 e2 u4 +888b47 e2 u4 + 905b48 e2 u4 + 927b49 e2 u4 + 944b50 e2 u4 + 966b51 e2 u4 +983b52 e2 u4 + 1005b53 e2 u4 + 1022b54 e2 u4 + 1044b55 e2 u4 + 1061b56 e2 u4 +1083b57 e2 u4 + 1100b58 e2 u4 + 1122b59 e2 u4 + 1139b60 e2 u4 + 1161b61 e2 u4 +1178b62 e2 u4 + 1200b63 e2 u4 + 1217b64 e2 u4 + 1239b65 e2 u4 + 1256b66 e2 u4 +1278b67 e2 u4 + 1295b68 e2 u4 + 1317b69 e2 u4 + 1334b70 e2 u4 + 1356b71 e2 u4 +1373b72 e2 u4 + 1395b73 e2 u4 + 1412b74 e2 u4 + 1434b75 e2 u4 + 1451b76 e2 u4 +1473b77 e2 u4 + 1490b78 e2 u4 + 1512b79 e2 u4 + 1529b80 e2 u4 + 1551b81 e2 u4 +1568b82 e2 u4 + 1590b83 e2 u4 + 1607b84 e2 u4 + 1629b85 e2 u4 + 9591b42 e4 u6 +9845b43 e4 u6 + 10084b44 e4 u6 + 10338b45 e4 u6 + 10577b46 e4 u6

Algebraically Independent Focus Quantities

125

+10831b47 e4 u6 + 11070b48 e4 u6 + 11324b49 e4 u6 + 11563b50 e4 u6 +11817b51 e4 u6 + 12056b52 e4 u6 + 12310b53 e4 u6 + 12549b54 e4 u6 +12803b55 e4 u6 + 13042b56 e4 u6 + 13296b57 e4 u6 + 13535b58 e4 u6 +13789b59 e4 u6 + 14028b60 e4 u6 + 14282b61 e4 u6 + 14521b62 e4 u6 +14775b63 e4 u6 + 15014b64 e4 u6 + 15268b65 e4 u6 + 15507b66 e4 u6 +15761b67 e4 u6 + 16000b68 e4 u6 + 16254b69 e4 u6 + 16493b70 e4 u6 +16747b71 e4 u6 + 16986b72 e4 u6 + 17240b73 e4 u6 + 17479b74 e4 u6 +17733b75 e4 u6 + 17972b76 e4 u6 + 18226b77 e4 u6 + 18465b78 e4 u6 +18719b79 e4 u6 + 18958b80 e4 u6 + 19212b81 e4 u6 + 19451b82 e4 u6 +19705b83 e4 u6 + 19944b84 e4 u6 + 20198b85 e4 u6 + 7110b28 e4 u8 +7393b29 e4 u8 + 7691b30 e4 u8 + 7974b31 e4 u8 + 8272b32 e4 u8 +8555b33 e4 u8 + 8853b34 e4 u8 + 9136b35 e4 u8 + 9434b36 e4 u8 +9717b37 e4 u8 + 10015b38 e4 u8 + 10298b39 e4 u8 + 10596b40 e4 u8 +10879b41 e4 u8 + 11177b42 e4 u8 + 11460b43 e4 u8 + 11758b44 e4 u8 +12041b45 e4 u8 + 12339b46 e4 u8 + 12622b47 e4 u8 + 12920b48 e4 u8 +13203b49 e4 u8 + 13501b50 e4 u8 + 13784b51 e4 u8 + 14082b52 e4 u8 +14365b53 e4 u8 + 14663b54 e4 u8 + 14946b55 e4 u8 + 15244b56 e4 u8 +15527b57 e4 u8 + 15825b58 e4 u8 + 16108b59 e4 u8 + 16406b60 e4 u8 +16689b61 e4 u8 + 16987b62 e4 u8 + 17270b63 e4 u8 + 17568b64 e4 u8 +17851b65 e4 u8 + 18149b66 e4 u8 + 18432b67 e4 u8 + 18730b68 e4 u8 +19013b69 e4 u8 + 19311b70 e4 u8 + 19594b71 e4 u8 + 19892b72 e4 u8 +20175b73 e4 u8 + 20473b74 e4 u8 + 20756b75 e4 u8 + 21054b76 e4 u8 +21337b77 e4 u8 + 21635b78 e4 u8 + 21918b79 e4 u8 + 22216b80 e4 u8 +22499b81 e4 u8 + 22797b82 e4 u8 + 23080b83 e4 u8 + 23378b84 e4 u8 +23661b85 e4 u8 + 137561b85 e6 u8 + ... .  (t) of the algebra S1 ,4 will have the Hence, the ordinary Hilbert series HS1,4 form (the first 100 terms):

HS1 ,4 (t) = 1 + t + 2t2 + 2t3 + 3t4 + 3t5 + 4t6 + 4t7 + 5t8 + 5t9 + 6t10 +6t11 + 7t12 + 7t13 + 8t14 + 8t15 + 9t16 + 9t17 + 10t18 + 10t19 + 253t20 +275t21 + 293t22 + 315t23 + 333t24 + 355t25 + 373t26 + 395t27 + 413t28 +435t29 + 453t30 + 475t31 + 493t32 + 515t33 + 533t34 + 555t35 + 573t36 +595t37 + 613t38 + 635t39 + 7763t40 + 8068t41 + 8384t42 + 8689t43 +9005t44 + 9310t45 + 9626t46 + 9931t47 + 10247t48 + 10552t49

126

The Center and Focus Problem

+10868t50 + 11173t51 + 21080t52 + 21639t53 + 22194t54 + 22753t55 +23308t56 + 23867t57 + 24422t58 + 24981t59 + 25536t60 + 26095t61 +26650t62 + 27209t63 + 27764t64 + 28323t65 + 28878t66 + 29437t67 +29992t68 + 30551t69 + 31106t70 + 31665t71 + 32220t72 + 32779t73 +33334t74 + 33893t75 + 34448t76 + 35007t77 + 35562t78 + 36121t79 (6.108) +36676t80 + 37235t81 + 37790t82 + 38349t83 + 38904t84 + 39463t85 +39974t86 + 40533t87 + 41087t88 + 41646t89 + 42200t90 + 42759t91 +41667t92 + 42204t93 + 42741t94 + 43278t95 + 23378t96 + 23661t97 +137561t99 + ... . We consider the first 100 terms in the expansion of Hilbert series of the algebra SI1,4 , which according to [33] is obtained from (4.31)–(4.32) in the following way: HSI1,4 (t) = H(S1,4 , 0, t, t) = 1 + t + 2t2 + 5t3 + 14t4 + 26t5 + 57t6 +119t7 + 248t8 + 461t9 + 864t10 + 1547t11 + 2737t12 + 4601t13 +7662t14 + 12383t15 + 19768t16 + 30664t17 + 47066t18 + 70770t19 +105300t20 + 153783t21 + 222506t22 + 317223t23 + 448337t24 +625302t25 + 865296t26 + 1184226t27 + 1609007t28 + 2164498t29 +2892657t30 + 3832653t31 + 5047384t32 + 6595561t33 + 8570829t34 +11061230t35 + 14202137t36 + 18120878t37 + 23011677t38 + 29058179t39 +36532673t40 + 45692819t41 + 56917559t42 + 70566839t43 +87158250t44 + 107183955t45 + 131345992t46 + 160313871t47 +195025339t48 + 236375592t49 + 285608968t50 + 343916178t51 +412927382t52 + 494204023t53 + 589868721t54 + 701958526t55 +833207339t56 + 986241378t57 + 1164562071t58 +1371538331t59 + 1611612886t60 + 1889064095t61 +2209499727t62 + 2578327522t63 + 3002568564t64 +3488999055t65 + 4046367551t66 + 4683127424t67 +5410102263t68 + 6237758509t69 + 7179427892t70 +8248015477t71 + 9459839180t72 + 10830705810t73 +12380506870t74 + 14128528398t75 + 16098882290t76 +18314961754t77 + 20805884383t78 + 23599922669t79 +26732062272t80 + 30236294034t81 + 34154507484t82 +38527430455t83 + 43404991223t84 + 48835760568t85 +54879055119t86 + 61592616546t87 + 69046625211t88

Algebraically Independent Focus Quantities

127

+77309433488t89 + 86463824763t90 + 96590473934t91 +107786664234t92 + 120147246938t93 + 133786223574t94 +148814805866t95 + 165366127962t96 + 183570124286t97

(6.109)

+203581864473t98 + 225552766408t99 + ... . Since for series (6.108) and (6.109), the following inequality holds HS1 ,4 (t) ≤ HSI1,4 (t), then in assumption that this inequality holds for remaining terms of the considered series, we obtain the inequality (S1 ,4 ) ≤ (SI1,4 ). Note that S1 ,4 is not a subalgebra in SI1,4 . Since from Theorem 4.9, we have (SI1,4 ) = 11, then according to the last inequality, we obtain that the following hypothesis can be true Hypothesis 6.1. Maximal number of algebraically independent generalized focus pseudo-quantities (as well as focus quantities) for differential system (6.77), which take part in solving the center and focus problem for given differential system, may not exceed 11.

6.5

On the Upper Bound of the Number of Algebraically Independent Focus Pseudo-Quantities for the Differential System s(1, 5)

Consider the differential system s(1, 5), which we write in the form x˙ = cx + dy + gx5 + 5hx4 y + 10kx3 y 2 + 10lx2 y 3 + 5mxy 4 + ny 5 , y˙ = ex + f y + px5 + 5qx4 y + 10rx3 y 2 + 10sx2 y 3 + 5uxy 4 + vy 5

(6.110)

with a finitely defined graded algebra of unimodular comitants S1,5 [33,34]. For this system, we write function (5.13) in the form U = k2 + a0 x3 + 3a1 x2 y + 3a2 xy 2 + a3 y 3 + b0 x4 + 4b1 x3 y + 6b2 x2 y 2 +4b3 xy 3 + b4 y 4 + c0 x5 + 5c1 x4 y + 10c2 x3 y 2 + 10c3 x2 y 3 + 5c4 xy 4 +c5 y 5 + d0 x6 + 6d1 x5 y + 15d2 x4 y 2 + 20d3 x3 y 3 + 15d4 x2 y 4 + 6d5 xy 5 +d6 y 6 + e0 x7 + 7e1 x6 y + 21e2 x5 y 2 + 35e3 x4 y 3 + 21e5 x2 y 5 + 7e6 xy 6 +e7 y 7 + f0 x8 + 8f1 x7 y + 28f2 x6 y 2 + 56f3 x5 y 3 + 70f4 y 4 +56f5 x3 y 5 + 28f6 x2 y 6 + 8f7 xy 7 + f8 y 8 + ..., (6.111)

128

The Center and Focus Problem

where k2 ≡ 0 is from (5.8), and a0 , a1 , ..., f7 , f8 , ... are unknown coefficients. Identity (5.12) along the trajectories of differential system (6.110) with function (6.111) splits into the following systems of equations (equality (5.15) is omitted): x3 : 3a0 c + 3a1 e = 0, x2 y : 6a1 c + 3a0 d + 6a2 e + 3a1 f = 0, xy 2 : 3a2 c + 6a1 d + 3a3 e + 6a2 f = 0,

(6.112)

y 3 : 3a2 d + 3a3 f = 0; x4 : 4b0 c + 4b1 e − e2 G1 = 0, x3 y : 12b1 c + 4b0 d + 12b2 e + 4b1 f + 2ceG1 − 2ef G1 = 0, x2 y 2 : 12b2 c + 12b1 d + 12b3 e + 12b2 f − c2 G1 + 2deG1 + 2cf G1 − f 2 G1 = 0,

(6.113)

xy 3 : 4b3 c + 12b2 d + 4b4 e + 12b3 f − 2cdG1 + 2df G1 = 0, y 4 : 4b3 d + 4b4 f − d2 G1 = 0; x5 : 5cc0 + 5c1 e = 0, x4 y : 20cc1 + 5c0 d + 20c2 e + 5c1 f = 0, x3 y 2 : 30cc2 + 20c1 d + 30c3 e + 20c2 f = 0, x2 y 3 : 20cc3 + 30c2 d + 20c4 e + 30c3 f = 0, xy 4 : 5cc4 + 20c3 d + 5c5 e + 20c4 f = 0, y 5 : 5c4 d + 5c5 f = 0; x6 : 6cd0 + 6d1 e − 2eg + e3 G2 + cp − f p = 0, x5 y : 6dd0 + 30cd1 + 30d2 e + 6d1 f + cg − f g − 3ce2 G2 + 3e2 f G2 − 10eh + 2dp + 5cq − 5f q = 0, x4 y 2 : 330dd1 + 60cd2 + 60d3 e + 30d2 f + 3c2 eG2 − 3de2 G2 − 6cef G2 + 3ef 2 G2 + 5ch − 5f h − 20ek + 10dq + 10cr − 10f r = 0, x3 y 3 : 60dd2 + 60cd3 + 60d4 e + 60d3 f − c3 G2 + 6cdeG2 + 3c2 f G2 − 6def G2 − 3cf 2 G2 + f 3 G2 + 10ck − 10f k − 20el + 20dr + 10cs − 10f s = 0, x2 y 4 : 660dd3 + 30cd4 + 30d5 e + 60d4 f − 3c2 dG2 + 3d2 eG2 + 6cdf G2 − 3df 2 G2 + 10cl − 10f l − 10em + 20ds + 5cu − 5f u = 0,

(6.114)

Algebraically Independent Focus Quantities xy 5 : 30dd4 + 6cd5 + 6d6 e + 30d5 f − 3cd2 G2 + 3d2 f G2 + 5cm − 5f m − 2en + 10du + cv − f v = 0,

129 (6.115)

y : 6dd5 + 6d6 f − d G2 + cn − f n + 2dv = 0; 6

3

x7 : 7ce0 + 7ee1 + 3a0 g + 3a1 p = 0, x6 y : 7de0 + 42ce1 + 42ee2 + 7e1 f + 6a1 g + 15a0 h + 6a2 p + 15a1 q = 0, x5 y 2 : 42de1 + 105ce2 + 105ee3 + 42e2 f + 3a2 g + 30a1 h + 30a0 k + 3a3 p + 30a2 q + 30a1 r = 0, x4 y 3 : 105de2 + 140ce3 + 140ee4 + 105e3 f + 15a2 h + 60a1 k + 30a0 l + 15a3 q + 60a2 r + 30a1 s = 0, x3 y 4 : 140de3 + 105ce4 + 105ee5 + 140e4 f + 30a2 k + 60a1 l + 15a0 m + 30a3 r + 60a2 s + 15a1 u = 0,

(6.116)

x2 y 5 : 105de4 + 42ce5 + 42ee6 + 105e5 f + 30a2 l + 30a1 m + 3a0 n + 30a3 s + 30a2 u + 3a1 v = 0, xy 6 : 42de5 + 7ce6 + 7ee7 + 42e6 f + 15a2 m + 6a1 n + 15a3 u + 6a2 v = 0, y 7 : 7de6 + 7e7 f + 3a2 n + 3a3 v = 0; x8 : 8cf0 + 8ef1 + 4b0 g + 4b1 p − e4 G3 = 0, x7 y : 8df0 + 56cf1 + 8f f1 + 56ef2 + 12b1 g + 20b0 h + 12b2 p + 20b1 q + 4ce3 G3 − 4e3 f G3 = 0, x6 y 2 : 56df1 + 168cf2 + 56f f2 + 168ef3 + 12b2 g + 60b1 h + 40b0 k + 12b3 p + 60b2 q + 40b1 r − 6c2 e2 G3 + 4de3 G3 + 12ce2 f G3 − 6e2 f 2 G3 = 0, x5 y 3 : 168df2 + 280cf3 + 168f f3 + 280ef4 + 4b3 g + 60b2 h + 120b1 k + 40b0 l + 4b4 p + 60b3 q + 120b2 r + 40b1 s + 4c3 eG3 − 12cde2 G3 − 12c2 ef G3 + 12de2 f G3 + 12cef 2 G3 − 4ef 3 G3 = 0, x4 y 4 : 280df3 + 280cf4 + 280f f4 + 280ef5 + 20b3 h + 120b2 k + 120b1 l + 20b0 m + 20b4 q + 120b3 r + 120b2 s + 20b1 u − c4 G3 + 12c2 deG3 − 6d2 e2 G3 + 4c3 f G3 − 24cdef G3 − 6c2 f 2 G3 + 12def 2 G3 + 4cf 3 G3 − f 4 G3 = 0, x3 y 5 : 280df4 + 168cf5 + 280f f5 + 168ef6 + 40b3 k + 120b2 l + 60b1 m + 4b0 n + 40b4 r + 120b3 s + 60b2 u + 4b1 v

130

The Center and Focus Problem − 4c3 dG3 + 12cd2 eG3 + 12c2 df G3 − 12d2 ef G3 − 12cdf 2 G3 + 4df 3 G3 = 0, x2 y 6 : 168df5 + 56cf6 + 168f f6 + 56ef7 + 40b3 l + 60b2 m + 12b1 n + 40b4 s + 60b3 u + 12b2 v − 6c2 d2 G3 + 4d3 eG3 + 12cd2 f G3 − 6d2 f 2 G3 = 0,

(6.117)

xy 7 : 56df6 + 8cf7 + 56f f7 + 8ef8 + 20b3 m + 12b2 n + 20b4 u + 12b3 v − 4cd3 G3 + 4d3 f G3 = 0, y 8 : 8df7 + 8f f8 + 4b3 n + 4b4 v − d4 G3 = 0. It is evident that linear systems of equations (6.112)–(6.117) with respect to the variables a0 , a1 , a2 , a3 , b0 , b1 ,...,b4 , c0 , c1 ,...,c5 , d0 , d1 ,...,d6 , e0 , e1 ,...,e7 , f0 , f1 ,...,f8 ,..., G1 , G2 , G3 ,... can be considered as a single system, which can be extended by adding, after the last equation from (6.117), an infinite number of equations, obtained from the equality of coefficients xα y β for α + β > 8 in identity (5.12). For obtaining the quantity G1 , we write system (6.113) in the matrix form A 1 B 1 = C1 ,

(6.118)

where ⎛ ⎜ ⎜ ⎜ A1 = ⎜ ⎝

4c 4d 0 0 0

4e 12c + 4f 12d 0 0

0 12e 12c + 12f 12d 0 ⎛ ⎜ ⎜ ⎜ B1 = ⎜ ⎜ ⎜ ⎝

b0 b1 b2 b3 b4 G1

0 0 12e 4c + 12f 4d ⎞

0 0 0 4e 4f

⎛ 0 ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎟ , C1 = ⎜ 0 ⎜ ⎟ ⎟ ⎝ 0 ⎠ 0

−e2 2ce − 2ef 2de + 2cf − c2 − f 2 −2cd + 2df −d2

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟. ⎠

(6.119)

Since the dimension of the matrix A1 is 5 × 6, then it is clear that we have at least one free variable. Therefore, choosing one of bi (i ∈ {0, , 1, ..., 4}) as a free variable, using the Cramer’s rule for system (6.118), we obtain B1,i bi G1 = , (6.120) σ1,i for any fixed i, equal to 0, 1, 2, 3, 4, where B1,i , σ1,i are polynomials in coefficients of system (6.110), and bi are undetermined coefficients of the function U (x, y) from (6.111). We obtain that b0 = b1 = b2 = b3 = b4 = 0 can be

Algebraically Independent Focus Quantities

131

taken as one particular solution of system (6.113). This means that G1 can be considered equal to zero. System (6.115) consists of seven equations. Writing these equations in the matrix form A2 B2 = C2 (see, Appendix 8) and performing similar reasoning as in the abovementioned case, we obtain

G2 =

G2,i + D2,i di σ2,i

(6.121)

for each i = 0, 6. We are interested in the degree of polynomials G2,i with respect to the coefficients of the system s(1, 5) from (6.110). The indicated degree coincides with the degree of the Cramer determinant ΔG2 . So the degree of G2,i with respect to the coefficients of the system s(1, 5) from (6.110) will be degG2,i = 8 for all i = 0, 6. Considering the dimension of the system, we obtain that G2,i has type (0, 7, 1), i.e. G2 is a homogeneous polynomial of degree 7 with respect to the coefficients of the linear part and homogeneous of degree 1 with respect to the coefficients of the nonhomogeneity of the fifth order of the system s(1, 5) from (6.110). Zero in (0, 7, 1) shows that expressions G2,i does not contain phase variables x, y. Note that in addition to these results obtained from the analysis of system (6.115), computer calculations were carried out and explicit forms of polynomials G2,i from (6.121) were determined for each fixed i = 0, 6 (see, Appendix 9). It was determined that G2,i (i = 0, 6) are homogeneous polynomials with respect to the coefficients of the system s(1, 5) from (6.110), and at the same time, for i = 0, 1, 2, 3, 4, 5, 6, they are polynomials of isobarities with weights, respectively (5, −1), (4, 0), (3, 1), (2, 2), (1, 3), (0, 4), (−1, 5).

(6.122)

We also note that σ2,i are polynomials only on the coefficients c, d, e, f of the linear part of the system s(1, 5) from (6.110). In the future, we will need an explicit form of the operators X1 , ..., X4 of Lie algebra L4 for system (6.116), the expressions for which are obtained from §5:

X1 = x

∂ ∂ ∂ ∂ + D1 , X2 = y + D2 , X3 = x + D3 , X4 = y + D4 , ∂x ∂x ∂y ∂y (6.123)

132

The Center and Focus Problem

where ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ −e − 4g − 3h − 2k −l +n − 5p ∂d ∂e ∂g ∂h ∂k ∂l ∂n ∂p ∂ ∂ ∂ ∂ −4q − 3r − 2s −u , ∂q ∂r ∂s ∂u ∂ ∂ ∂ ∂ ∂ ∂ D2 = e + (f − c) −e +p + (q − g) + (r − 2h) ∂c ∂d ∂f ∂g ∂h ∂k ∂ ∂ ∂ ∂ ∂ ∂ +(s − 3k) + (u − 4l) + (v − 5m) −p − 2q − 3r ∂l ∂m ∂n ∂q ∂r ∂s ∂ ∂ −4s − 5u , ∂u ∂v ∂ ∂ ∂ ∂ ∂ ∂ D3 = −d + (c − f ) +d − 5h − 4k − 3l ∂c ∂e ∂f ∂g ∂h ∂k ∂ ∂ ∂ ∂ ∂ −2m − n + (g − 5q) + (h − 4r) + (k − 3s) ∂l ∂m ∂p ∂q ∂r ∂ ∂ ∂ +(l − 2u) + (m − v) +n , ∂s ∂u ∂v D1 = d

∂ ∂ ∂ ∂ ∂ ∂ +e −h − 2k − 3l − 4m ∂d ∂e ∂h ∂k ∂l ∂m ∂ ∂ ∂ ∂ ∂ ∂ −5n +p −r − 2s − 3u − 4v . ∂n ∂p ∂r ∂s ∂u ∂v

D4 = −d

(6.124)

Using formula of comitant’s weight (3.3), for differential system (6.110), we obtain that numerators of fractions (6.121) can be coefficients in comitants of the weight −1 of type (6, 7, 1). Using Lie differential operator D3 from (6.124) for differential system (6.110), we obtain the system of six linear nonhomogeneous partial differential equations: D3 (G2,0 + D2,0 f0 ) = G2,1 + D2,1 d1 , D3 (G2,1 + D2,1 d1 ) = −G2,2 − D2,2 d2 , −D3 (G2,2 + D2,2 d2 ) = G2,3 + D2,3 d3 , D3 (G2,3 + D2,3 d3 ) = −G2,4 − D2,4 d4 , −D3 (G2,4 + D2,4 d4 ) = G2,5 + D2,5 d5 , D3 (G2,5 + D2,5 d5 ) = −G2,6 − D2,6 d6

(6.125)

with seven unknowns d0 , d1 , ..., d6 . According to Lemma 2.4, system (6.125) has an infinite number of solutions. Note that a particular solution of this system is d0 = d1 = ... = d6 = 0, for which the polynomial f6 (x, y) = G2,0 x6 − 6G2,1 x5 y − 3G2,2 x4 y 2 + 2G2,3 x3 y 3 +3G2,4 x2 y 4 − 6G2,5 xy 5 − G2,6 y 6

(6.126)

Algebraically Independent Focus Quantities

133

is a centro-affine comitant of differential system (6.110). This fact is also confirmed by Theorem 2.2 with the operators X1 − X4 from (6.123), (6.124) for differential system (6.110), for which X1 (f6 ) = X4 (f6 ) = f6 , X2 (f6 ) = X3 (f6 ) = 0. It is obvious that differential system (6.125) has an infinite number of solutions d0 , d1 , ..., d6 , which define centro-affine comitants of type (6, 7, 1), which are written as f6 (x, y) = (G2,0 + D2,0 d0 )x6 − 6(G2,1 + D2,1 d1 )x5 y − 3(G2,2 +D2,2 d2 )x4 y 2 + 2(G2,3 + D2,3 d3 )x3 y 3 + 3(G2,4 + D2,4 d4 )x2 y 4

(6.127)

−6(G2,5 + D2,5 d5 )xy − (G2,6 + D2,6 d6 )y . 5

6

According to the abovementioned, comitant (6.126) belongs to the linear space (6,7,1) S1,5 . Note that comitant (6.126) on the variety V from (5.9) for differential system (6.110) has the form f6 (x, y)|V = L2 (x2 + y 2 )3 (G2 |V = L2 ),

(6.128)

where L2 = 20(g + 2k + m + q + 2s + v), is the first nonzero Lyapunov quantity of differential system (6.110) on an invariant variety V. Consider the extension of system (6.112)–(6.117), which is obtained from identity (5.12) for differential system (6.110) and function (6.111), that contains the quantity G2k , which we write in the matrix form A2k B2k = C2k . We denote by mG2k the number of equations, and by nG2k the number of unknowns of this system. Note that this number is written as: mG2k = 2 · 2 + 3 +2 · 4 + 3 +2 · 6 + 3 +... + 2 · 2k + 3, ' () * ' ' '

G2

() G4

* ()

G6 . . .

* ()

*

G2k

for k = 1, 2, 3, . . . . There from we obtain mG2k = 2k 2 + 5k.

(6.129)

From these systems, it is obtained G2 k =

G2k,i1 ,i2 ,...,ik + B2k,i1 ,i2 ,...,ik bi1 + · · · + Z2k,i1 ,i2 ,...,ik zik . σ2k,i1 ,i2 ,...,ik

(6.130)

134

The Center and Focus Problem

It is important to determine the degree of the polynomial G2k,i1 ,i2 ,...,ik + +B2k,i1 ,i2 ,...,ik bi1 + · · · + Z2k,i1 ,i2 ,...,ik zik with respect to the coefficients of the differential system (6.110). Note that the degree of nonzero coefficient of the polynomial G2i (i = 1, k) with respect to the coefficients of system (6.110) in Cramer’s determinant of the order mG2k , when the coefficients in G2k are replaced with the free members of the considered system, forms the following diagram: G2 , G4 , G6 , ... , G2(k−1) , ↓ 7

↓ 18

↓

G2k .

↓

33 k(2k + 1) − 3

↓ 1

Then the degree of the polynomials G2k,i1 ,i2 ,...,ik + B2k,i1 ,i2 ,...,ik bi1 + · · · + Z2k,i1 ,i2 ,...,ik zik with respect to the coefficients of system (6.110), denoted by NG2k , will be written as NG2k = mG2k + 2

k(k − 1) + 1, 2

from where we obtain NG2k = 3k 2 + 4k + 1.

(6.131)

It is the degree of homogeneity of polynomials G2k,i1 ,i2 ,...,ik + B2k,i1 ,i2 ,...,ik bi1 + · · · + Z2k,i1 ,i2 ,...,ik zik with respect to the coefficients of system (6.110), which are polynomials of type (0, d0 , d1 ), (6.132) where d0 is the degree of homogeneity of the polynomial with respect to the coefficients of the linear part, d1 is the degree of homogeneity of the polynomial with respect to the coefficients of the nonhomogeneity of the fifth order of the system s(1, 5) from (6.110). Since δ = 2(2k+1) and d1 = k, then d0 = NGk − k. Based on this, we find that a comitant of weight −1 of the system s(1, 5) from (6.110), which contains G2k,i1 ,i2 ,...,ik + +B2k,i1 ,i2 ,...,ik bi1 + · · · + Z2k,i1 ,i2 ,...,ik zik corresponding to the quantity G2k for k = 1, 2, 3, ... (Gn = 0 if n = 2k) as a semi-invariant, has type 

 2(2k + 1), 3k 2 + 3k + 1, k ,

(6.133)

where 2(2k+1) is the degree of homogeneity of the comitant in phase variables x, y; 3k 2 +3k+1 is the degree of homogeneity of the comitant in the coefficients of linear part; and k is the degree of homogeneity of the comitant in the coefficients of the nonhomogeneity of the fifth order of the system s(1, 5) from (6.110). Consider the differential system s(1, 5) from (6.110). In this case, according to the paper [33] using Theorem 4.10 the following is true:

Algebraically Independent Focus Quantities

135

Theorem 6.7. Dimension of linear space of centro-affine comitants of type (d) = (δ, d0 , d1 ) for the differential system s(1, 5) from (6.110), denoted by (d) dimR V1,5 , is equal to the coefficient of the monomial uδ bd0 f d1 in decomposition of generalized Hilbert series from (4.42)–(4.43) for Sibirsky algebra S1,5 of comitants of the considered system. Consider the subalgebra S1 ,5 ⊂ S1,5 , which we write in the form # (d ) S1 ,5 = S1,5 , (6.134) (d) (d )

where by S1,5 the linear spaces are denoted: (0,0,0)

S1,5

(0,1,0)

= R, S1,5

2(2k+1),3k2 +3k+1,k

, ..., S1,5

, k = 1, 2, ...,

(6.135)

as well as spaces from S1,5 , which contain all kinds of their products. Since algebra S1 ,5 is a graded subalgebra in a finitely defined algebra S1,5 , then according to Proposition 4.1, we obtain (S1 ,5 ) ≤ (S1,5 ). From this inequality and from the fact that (S1,5 ) = 15 (see Theorem 4.12), according to Remark 2.3 on semi-invariants and the fact that generalized focus pseudoquantities are coefficients of some comitants, we have that the following takes place: Theorem 6.8. Maximal number of algebraically independent generalized focus pseudo-quantities in the center and focus problem for differential system (6.110) does not exceed 15. According to Proposition 4.2, Remark 5.1, and equality (5.14), it follows that the maximal number of algebraically independent focus quantities Lk (k = 1, ∞) cannot exceed maximal number of algebraically independent generalized focus pseudo-quantities G2k,i1 ,i2 ,...,ik + +B2k,i1 ,i2 ,...,ik bi1 + · · · + Z2k,i1 ,i2 ,...,ik zik . There from using Theorem 6.8, we obtain Consequence 6.4. Maximal number of algebraically independent focus quantities that take part in solving the center and focus problem for differential system (6.110) does not exceed 15. The generalized generating function of space (6.134) can be written as: (0,0,0)

Φ(V1,5 , u, b, f ) = dimR S1,5 +

∞ 

(0,1,0)

+ dimR S1,5

b

(2(2k+1),3k2 +3k+1,k) 2(2k+1) 3k2 +3k+1 k

dimR S1,5

u

b

f .

(6.136)

k=1

Using computer, expand the Hilbert series H(S1,5 , u, b, f ) (24.7)–(24.9) in a power series. Then for (6.136), we have

from

Φ(V1,5 , u, b, f ) = 1 + b + 33u6 b7 f + 585u10 b19 f 2 + 5616u14 b37 f 3 + ... +C2(2k+1),3k2 +3k+1,k u2(2k+1) b(3k

2

+3k+1) k

f + ..., (6.137)

136

The Center and Focus Problem

where C2(k+1), 12 (15k2 +11k+2),2k is an undetermined coefficient. Using this generalized generating function and the Hilbert series H(S1,5 , u, b, f ) from Theorem 6.7, there were obtained the first terms up to u22 b91 f 5 (δ + d0 + d1 ≤ 118) in the Hilbert series H(S1 ,5 , u, b, f ):  , u, b, f ) = 1 + b + 2b2 + 2b3 + 3b4 + 3b5 + 4b6 + 4b7 + 5b8 + 5b9 H(S1,5

+6b10 + 6b11 + 7b12 + 7b13 + 8b14 + 8b15 + 9b16 + 9b17 + 10b18 + 10b19 +11b20 + 11b21 + 12b22 + 12b23 + 13b24 + 13b25 + 14b26 + 14b27 + 15b28 +15b29 + 16b30 + 16b31 + 17b32 + 17b33 + 18b34 + 18b35 + 19b36 + 19b37 +20b38 + 20b39 + 21b40 + 21b41 + 22b42 + 22b43 + 23b44 + 23b45 +24b46 + ... + u22 (46666b78 f 4 + 47358b79 f 4 + 48029b80 f 4 + 48721b81 f 4 +49392b82 f 4 + 50084b83 f 4 + 50755b84 f 4 + 51447b85 f 4 + 52118b86 f 4 +52810b87 f 4 + 53481b88 f 4 + 54173b89 f 4 + 54844b90 f 4 + 55536b91 f 4 +176322b91 f 5 ) + ...  Hence, the ordinary Hilbert series HS1,5 (t) of the algebra S1 ,5 will have the form (the first 119 terms):

 HS1,5 (t) = 1 + t + 2t2 + 2t3 + 3t4 + 3t5 + 4t6 + 4t7 + 5t8 + 5t9 + 6t10

+6t11 + 7t12 + 7t13 + 41t14 + 47t15 + 54t16 + 60t17 + 67t18 + 73t19 +80t20 + 86t21 + 93t22 + 99t23 + 106t24 + 112t25 + 119t26 + 125t27 +504t28 + 546t29 + 595t30 + 1222t31 + 1306t32 + 1389t33 + 1473t34 +1556t35 + 1640t36 + 1723t37 + 1807t38 + 1890t39 + 1974t40 + 2057t41 +4598t42 + 4863t43 + 5129t44 + 8915t45 + 9362t46 + 9808t47 +10255t48 + 10701t49 + 11148t50 + 11594t51 + 12041t52 +12487t53 + 18550t54 + 19175t55 + 19801t56 + 20426t57 +21052t58 + 37686t59 + 38983t60 + 40300t61 + 61616t62 + 63602t63 +65589t64 + 67575t65 + 69562t66 + 71548t67 + 73535t68 + 75521t69 +77508t70 + 79494t71 + 81481t72 + 83467t73 + 85454t74 + 87440t75 +89427t76 + 91413t77 + 93400t78 + 95386t79 + 97373t80 + 99359t81 +101346t82 + 139347t83 + 141998t84 + 144669t85 + 147320t86 +149991t87 + 152642t88 + 155313t89 + 157964t90 + 160635t91 +163286t92 + 165957t93 + 168608t94 + 171279t95 + 173930t96 +176601t97 + 179252t98 + 181380t99 + 184025t100 + 186690t101 +189335t102 + 192000t103 + 191289t104 + 193913t105 + 193109t106 +195697t107 + 198265t108 + 185392t109 + 187781t110 + 174723t111

Algebraically Independent Focus Quantities +176931t112 + 163780t113 + 108892t114 + 110253t115 +54903t116 + 55595t117 + 176382t118 + ...

137 (6.138)

We consider the first 119 terms in the expansion of Hilbert series of the algebra SI1,5 from (4.42)–(4.43). Replacing z = t, we obtain HSI1,5 (t) = H(S1,5 , 0, t, t) = 1 + t + 4t2 + 8t3 + 26t4 + 53t5 + 146t6 +305t7 + 704t8 + 1417t9 + 2920t10 + 5533t11 + 10500t12 + 18825t13 +33444t14 + 57120t15 + 96303t16 + 157599t17 + 254508t18 + 401472t19 +625182t20 + 955251t21 + 1442076t22 + 2142840t23 + 3149178t24 +4566267t25 + 6554694t26 + 9300484t27 + 13076140t28 + 18198949t29 +25118690t30 + 34359893t31 + 46645739t32 + 62820314t33 + 84019460t34 +111568250t35 + 147213784t36 + 192990661t37 + 251534302t38 +325907859t39 + 420016674t40 + 538389135t41 + 686719824t42 +871593216t43 + 1101188574t44 + 1384936842t45 + 1734423882t46 +2162969685t47 + 2686776843t48 + 3324416523t49 + 4098277602t50 +5033946165t51 + 6162015960t52 + 7517347113t53 + 9141313732t54 +11080921339t55 + 13391579524t56 + 16136061599t57 +19387898270t58 + 23230161917t59 + 27759598166t60 +33085209860t61 + 39333260630t62 + 46645639450t63 +55185881485t64 + 65137293814t65 + 76710167634t66 +90139636710t67 + 105694278048t68 + 123673683567t69 +144418662324t70 + 168308506209t71 + 195773044560t72 +227289574704t73 + 263397050880t74 + 304692662715t75 +351848461524t76 + 405607571979t77 + 466803608430t78 +536356493511t79 + 615295057113t80 + 704752453114t81 +805992362230t82 + 920404001941t83 + 1049532417158t84 +1195073066570t85 + 1358906736914t86 + 1543093711865t87 +1749913882256t88 + 1981860511040t89 + 2241686191348t90 +2532396208270t91 + 2857301061364t92 + 3220009454859t93 +3624488148738t94 + 1981860511040t89 + 2241686191348t90 +2532396208270t91 + 2857301061364t92 + 3220009454859t93 +3624488148738t94 + 4075054619802t95 + 4576445103411t96 +5133806955846t97 + 5752775830170t98 + 6439467809532t99 +7200566719746t100 + 8043316119915t101 + 8975617875264t102

138

The Center and Focus Problem

+10006024079634t103 + 11143848111366t104 + 12399156613041t105 +13782894327786t106 + 15306876284184t107 + 16983927856168t108 +18827877291745t109 + 20853712561492t110 + 23077574432714t111 +25516931762351t112 + 28190575349780t113 + 31118813506166t114

(6.139)

+34323466911800t115 + 37828086439832t116 + 41657949323224t117 +45840301322554t118 + ... Since for series (6.138) and (6.139), the following inequality holds: HS1 ,5 (t) ≤ HSI1,5 (t), then in assumption that this inequality holds for remaining terms of the considered series, we obtain the inequality (S1 ,5 ) ≤ (SI1,5 ). Note that S1 ,5 is not a subalgebra in SI1,5 . Since from Theorem 4.12, we have (SI1,5 ) = 13, then according to the last inequality, we obtain that the following can be true: Hypothesis 6.2. Maximal number of algebraically independent generalized focus pseudo-quantities (as well as focus quantities) for differential system (6.110) that take part in solving the center and focus problem for the given differential system may not exceed 13.

6.6

Comitants that Have Generalized Focus Pseudo-Quantities of the System s(1, 2, 3) as Coefficients, and Their Sibirsky Graded Algebra

Consider the differential system s(1, 2, 3), which we write in the form x˙ = cx + dy + gx2 + 2hxy + kx2 + px3 + 3qx2 y + 3rxy 2 + sy 3 , y˙ = ex + f y + lx2 + 2mxy + ny 2 + tx3 + 3ux2 y + 3vxy 2 + wy 3

(6.140)

with a finitely defined graded algebra of unimodular comitants S1,2,3 . For this system, we write function (5.13) in the form U = k2 + a0 x3 + 3a1 x2 y + 3a2 xy 2 + a3 y 3 + b0 x4 + 4b1 x3 y + 6b2 x2 y 2 +4b3 xy 3 + b4 y 4 + c0 x5 + 5c1 x4 y + 10c2 x3 y 2 + 10c3 x2 y 3 + 5c4 xy 4 +c5 y 5 + d0 x6 + 6d1 x5 y + 15d2 x4 y 2 + 20d3 x3 y 3 + 15d4 x2 y 4 + 6d5 xy 5

Algebraically Independent Focus Quantities

139

+d6 y 6 + e0 x7 + 7e1 x6 y + 21e2 x5 y 2 + 35e3 x4 y 3 + 21e5 x2 y 5 + 7e6 xy 6 +e7 y 7 + f0 x8 + 8f1 x7 y + 28f2 x6 y 2 + 56f3 x5 y 3 + 70f4 y 4 +56f5 x3 y 5 + 28f6 x2 y 6 + 8f7 xy 7 + f8 y 8 + ..., (6.141) where k2 ≡ 0 is from (5.8), and a0 , a1 , ..., f7 , f8 , ... are unknown coefficients. Identity (5.12) along the trajectories of differential system (6.140) with function (6.141) splits into the following systems of equations with respect to the variables a0 , a1 ,..., f7 , f8 , G1 , G2 , G3 , ... (equality (5.15) is omitted). For obtaining the quantity G1 , we write equations, in which the identity (5.12) decomposes in the case of differential system (6.140) into the matrix form 1 , 1 B 1 = C A where

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜  ⎜ A1 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

3c 3d 0 0 3g 6h 3k 0 0

3e 6c+3f 6d 0 3l 6g+6m 12h+3n 6k 0 0 0 0 0 0 0 12e 4c+12f 4d

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 = ⎜ B ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

a0 a1 a2 a3 b0 b1 b2 b3 b4 G1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 6e 3c+6f 3d 0 6l 3g+12m 6h+6n 3k 0 0 0 0 0 0 0 4e 4f ⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 1 = ⎜ C ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 3e 3f 0 0 3l 6m 3n

(6.142)

0 0 0 0 4c 4d 0 0 0

0 0 0 0 4e 12c+4f 12d 0 0 ⎞

0 0 0 0 −e2 2ce−2ef −c2 +2de+2cf − f 2 −2cd+2df −d2

0 0 0 0 0 12e 12c+12f 12d 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

2eg − cl + f l −cg + f g + 4eh − 2dl − 2cm + 2f m −2ch + 2f h + 2ek − 4dm − cn + f n −ck + f k − 2dn 2ep − ct + f t −cp + f p + 6eq − 2dt − 3cu + 3f u −3cq + 3f q + 6er − 6du − 3cv + 3f v −3cr + 3f r + 2es − 6dv − cw + f w −cs + f s − 2dw

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠ (6.143)

140

The Center and Focus Problem

For each fixed i ∈ {0, 1, ..., 4} using Cramer’s rule from system (6.142), we obtain 1,i bi  1,i + B G (6.144) G1 = , σ 1,i  1,i , B 1,i , σ where G 1,i are polynomials in coefficients of differential system (6.140), and bi are undetermined coefficients of the function U (x, y) from (6.141). Studying matrix equation (6.142) for differential system (6.140), we find  1,i for any fixed i from (6.144) can be written that the focus pseudo-quantity G as  + G   , (i = 0, 1, 2, 3, 4),  1,i = G (6.145) G 1,i 1,i   (G   respectively) are homogeneous polynomials of degree 8 (9 where G 1,i 1,i respectively) with respect to the coefficients of the linear part and homogeneous of degree 2 with respect to the coefficients of the quadratic part (of degree 1 with respect to the coefficients of the cubic part, respectively) of differential system (6.140). From (1.40)–(1.41), (1.44)–(1.45), (1.47)–(1.48), (1.50)–(1.51) operators of Lie algebra L4 for differential system (6.140) are obtained: ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +d −e −g +k − 2l − m − 2p −q ∂x ∂d ∂e ∂g ∂k ∂l ∂m ∂p ∂q ∂ ∂ ∂ ∂ +s − 3t − 2u −v , ∂s ∂t ∂u ∂v ∂ ∂ ∂ ∂ ∂ ∂ + e + (f − c) −e +l + (m − g) + (n X2 = y ∂x ∂c ∂d ∂f ∂g ∂h ∂ ∂ ∂ ∂ ∂ ∂ −2h) −l − 2m +t + (u − p) + (v − 2q) + (w ∂k ∂m ∂n ∂p ∂q ∂r ∂ ∂ ∂ ∂ −3r) −t − 2u − 3v , ∂s ∂u ∂v ∂w ∂ ∂ ∂ ∂ ∂ ∂ ∂ X3 = x − d + (c − f ) +d − 2h −k + (g − 2m) ∂y ∂c ∂e ∂f ∂g ∂h ∂l ∂ ∂ ∂ ∂ ∂ ∂ +(h − n) +k − 3q − 2r −s + (p − 3u) + (q ∂m ∂n ∂p ∂q ∂r ∂t ∂ ∂ ∂ −2v) + (r − w) +s , ∂u ∂v ∂w ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ X4 = y −d +e −h − 2k +l −n −q − 2r ∂y ∂d ∂e ∂h ∂k ∂l ∂n ∂q ∂r ∂ ∂ ∂ ∂ −3s +t −v − 2w . ∂s ∂t ∂v ∂w Applying these operators to expressions from (6.145), we find the equalities X1 = x

X1 (f4 ) = X4 (f4 ) = f4 , X2 (f4 ) = X3 (f4 ) = 0, X1 (f ) = X4 (f ) = f , X2 (f ) = X3 (f ) = 0, 4

4

4

4

4

Algebraically Independent Focus Quantities

141

where   x4 − 4G   x3 y + 2G   x2 y 2 + 4G   xy 3 − G  1,4 y 4 , f4 (x, y) = G 1,0 1,1 1,2 1,3   x4 − 4G   x3 y + 2G   x2 y 2 + 4G   xy 3 − G   y 4 f4 (x, y) = G 1,3 1,4 1,0 1,1 1 ,2

(6.146)

  are  , G are comitants of weight −1 of differential system (6.140), and G 1,i 1,i from (6.145). According to the abovementioned and (3.1), comitants (6.146) belong to the linear spaces (4,8,2,0) (4,9,0,1) (6.147) S1,2,3 , S1,2,3 , which are components of graded Sibirsky algebra of comitants S1,2,3 for differential system (6.140). According to (6.144), for example, for bi = 0 (i = 0, 4) on the variety V from (5.11) for (6.145), (6.146), we find that between the first focus quantity L1 of differential system (6.140) and comitants (6.146), the following equality holds:  f4 (x, y) + f4 (x, y) | V = 8L1 (x2 + y 2 )2 (G1 |V = 8L1 ), where L1 =

1 {2[g(l − h) − k(h + n) + m(l + n)] − 3[p + r + u + w]} . 4

If you exclude the numerical constant and consider the notation of coefficients of this differential system, then the last expression coincides with the focus quantity of this system from [10, p. 25]. For obtaining the quantity G2 for differential system (6.140) from identity (5.12), similarly, we obtain the following matrix equation (see Appendix 10): 2 . 2 B 2 = C A

(6.148)

For any fixed i ∈ {0, 1, ..., 4}, j ∈ {0, 1, 2, ..., 6}, we find the expression G2 =

 2,i,j + B 2,i,j bi + D  2,i,j dj G . σ 2,i,j

(6.149)

Studying matrix equation (6.148), we find that focus pseudo-quantity from (6.149) can be written as a homogeneity of degree 24 and presented as       2,i,j = G G 2,i,j + G2,i,j + G2,i,j ,

(6.150)

 , G     where G 2,i,j 2,i,j and G2,i,j are homogeneities of a similar type from (3.1), i.e. of the form (0, d0 , d1 , d2 ). Hence, we have the formulas (0, 20, 4, 0), (0, 21, 2, 1) and (0, 22, 0, 2), respectively. Note that on the variety V from (5.11)  2,2,j (j = 0, 6) have the form for differential system (6.140) quantities G  2,2,j |V = 2304L2 (j = 0, 2, 4, 6), G  2,2,j |V = 0, (j = 1, 3, 5). G

142

The Center and Focus Problem

On the other hand, the second focus quantity L2 of differential system (6.140) can be written using an expression from (6.150) as follows:   |V + G   |V + G   24L2 = G 2,2,j 2,2,j 2,2,j |V (j = 0, 2, 4, 6), where   2,2,j |V = 4(62g 3 h − 2gh3 + 95g 2 hk − 2h3 k + 38ghk 2 + 5hk 3 − 62g 3 l G

+27gh2 l − 39g 2 kl + 29h2 kl − 15gk 2 l − 8ghl2 + 15hkl2 − 5gl3 + 53g 2 hm +66ghkm + 13hk 2 m − 127g 2 lm − 6h2 lm − 68gklm − 15k 2 lm − 13hl2 m −5l3 m + 6ghm2 + 6hkm2 − 63glm2 − 29klm2 + 2lm3 + 6g 3 n + 61gh2 n +72g 2 kn + 63h2 kn + 33gk 2 n + 5k 3 n − 10ghln + 68hkln − 33gl2 n +15kl2 n − 72g 2 mn − 6h2 mn + 10gkmn + 8k 2 mn − 66hlmn − 38l2 mn −61gm2 n − 27km2 n + 2m3 n + 72ghn2 + 127hkn2 − 72gln2 + 39kln2 −53hmn2 − 95lmn2 − 6gn3 + 62kn3 − 62mn3 ),   |V = −2(186g 2 p + 10h2 p + 117gkp + 45k 2 p + 59hlp + 15l2 p G 2,2,j +159gmp + 75kmp + 18m2 p + 143hnp + 89lnp + 196n2 p − 69ghq −57hkq + 69glq + 12klq + 9lmq + 60gnq + 3knq + 21mnq + 168g 2 r −6h2 r + 69gkr + 15k 2 r + 87hlr + 45l2 r + 123gmr + 39kmr + 18m2 r +171hnr + 129lnr + 222n2 r − 13ghs − 17hks − 15gls − 16hms − 15lms −16gns − 17kns − 19mns − 19ght − 15hkt − 17glt − 16hmt − 17lmt −16gnt − 15knt − 13mnt + 222g 2 u + 18h2 u + 129gku + 45k 2 u + 39hlu +15l2 u + 171gmu + 87kmu − 6m2 u + 123hnu + 69lnu + 168n2 u + 21ghv +9hkv + 3glv + 12klv − 57lmv + 60gnv + 69knv − 69mnv + 196g 2 w +18h2 w + 89gkw + 15k 2 w + 75hlw + 45l2 w + 143gmw + 59kmw +10m2 w + 159hnw + 117lnw + 186n2 w),   |V = −9(11pq + 15qr − 5ps − rs + pt + 5rt + 3qu − 5su + tu − 7pv G 2,2,j −3rv − 15uv + 7qw − sw + 5tw − 11vw). Choose a comitant of weight −1 of the differential system s(1, 2, 3) from  2,i,j + B 2,i,j bi + D  2,i,j dj as semi(6.140), which contains the expression G invariant. According to decomposition (6.150) and the types arising from it, we find that this comitant is the sum of the comitants, which belong to the linear spaces (6,20,4,0) (6,21,2,1) (6,22,0,2) S1,2,3 , S1,2,3 , S1,2,3 . (6.151) Using the same process and the matrix equation 3 B 3 = C 3 A

Algebraically Independent Focus Quantities

143

for any fixed i ∈ {0, 1, ..., 4}, j ∈ {0, 1, ..., 6}, k ∈ {0, 1, ..., 8} we obtain G3 =

3,i,j,k bi + D  3,i,j,k dj + F3,i,j,k fj  3,i,j,k + B G . σ 3,i,j,k

(6.152)

 3,i,j,k Similarly to the previous case, we find that the focus pseudo-quantity G is decomposed into a sum of four members of the same degree 43 with respect to the coefficients of differential system (6.140), which according to (3.1) will be written as (0, d0 , d1 , d2 ) and belong to the types (0, 37, 6, 0), (0, 38, 4, 1), (0, 39, 2, 2) and (0, 40, 0, 3). In this case, it follows that a comitant of the weight −1, which has one of the numerators of expression (6.152) as a semiinvariant, consists of the sum of comitants of system (6.140), which belong to the linear spaces (8,37,6,0)

S1,2,3

(8,38,4,1)

, S1,2,3

(8,39,2,2)

, S1,2,3

(8,40,0,3)

, S1,2,3

.

(6.153)

Following this process, we obtain a series of linear spaces (6.147), (6.151), (6.153) etc. of the comitants of system (6.140). Note that the corresponding generalized focus pseudo-quantities Gk of the given system are the sum of the coefficients of these comitants. Similarly, it is not difficult to derive a general formula of comitants that have generalized focus pseudo-quantities corresponding to the Gk as coefficients, which decompose into sum of comitants of differential system (6.140) of the corresponding types 

1 2(k + 1), (5k 2 + 9k + 2) + i, 2(k − i), i 2

(i = 0, k).

Consider the subalgebra S1 ,2,3 ⊂ S1,2,3 , which we write in the form S1 ,2,3 =

#

(d )

S1,2,3 ,

(6.154)

(d) (d )

where by S1,2,3 the linear spaces are denoted: (0,0,0,0)

S1,2,3

(0,1,0,0)

= R, S1,2,3

(2(k+1), 12 (5k2 +9k+2)+i,2(k−i),i)

, ..., S1,2,3

(6.155)

(i = 0, k, k = 1, 2, ...) as well as spaces from S1,2,3 , which contain all kinds of their products. Since the algebra S1 ,2,3 is a graded subalgebra in a finitely defined algebra S1,2,3 , then according to Proposition 4.1, we obtain (S1 ,2,3 ) ≤ (S1,2,3 ). From this inequality and from the fact that using formula (2.54), in which for differential system (6.140) we have m0 = 1, m1 = 2, m2 = 3, we obtain (S1,2,3 ) = 17.

144

The Center and Focus Problem

Then according to Remark 2.3 on semi-invariants and the fact that generalized focus pseudo-quantities are coefficients of some comitants of the given differential system, we obtain that the following is true: Theorem 6.9. Maximal number of algebraically independent generalized focus pseudo-quantities in the center and focus problem for differential system (6.140) does not exceed 17. According to Proposition 4.2, Observation 5.2, and equality (5.14), it follows that maximal number of algebraically independent focus quantities Lk (k = 1, ∞) cannot exceed maximal number of algebraically independent generalized focus pseudo-quantities. There from using Theorem 6.10, we obtain Consequence 6.5. Maximal number of algebraically independent focus quantities that take part in solving the center and focus problem for differential system (6.140) does not exceed 17.

6.7

On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Differential System s(1, m1 , ..., m )

Consider the differential system s(1, m1 , ..., m ) : dx  dy  = Qmi (x, y), = Pmi (x, y), dt dt i=0 i=0 



(6.156)

where Pmi and Qmi are homogeneous polynomials of degree mi ≥ 1 with respect to x and y, and m0 = 1. According to Section 5.1, the problem is to determine a majorant estimate of the number λ of algebraically independent elements from (5.1), (5.2), i.e. for any nontrivial polynomial with respect to  0, (λ ≤ ω) the variables Li1 , Li2 ,..., Liλ the inequality F (Li1 , Li2 , ..., Liλ ) = holds. Focus quantities of differential system (6.156) form an infinite sequence of polynomials from coefficients of this system (5.1). It is known from the paper [33], that differential system s(1, m1 , ..., m ) admits the group GL(2, R), to which the reductive Lie algebra L4 corresponds, that consists of operators (1.40)–(1.41), (1.44)–(1.45), (1.47)–(1.48), (1.50)–(1.51) of linear representation of this group in the space of phase variables and coefficients of this system. This algebra generates a graded Sibirsky algebra of invariant polynomials with respect to the

Algebraically Independent Focus Quantities

145

unimodular group SL(2, R) ⊂ GL(2, R), which we write in the form  (d) S1,m1 ,...,m = S1,m1 ,...,m , (6.157) (d) (d)

where (d) is called a type of the space S1,m1 ,...,m and has the form (3.1), which is a finite-dimensional linear space of invariant polynomials (homogeneous comitants, invariants of degree δ with respect to the phase variables x, y and of degree di with respect to the coefficients of the polynomial Pmi and Qmi of system (6.156)). Similarly to the considered examples in this chapter, it can be shown that to each focus quantity Lk (k = 1, ∞) one can associate finite-dimensional linear spaces of invariant polynomials (unimodular comitants [33]) (d(k) )

(6.158)

S1,m1 ,...,m (k = 1, 2, ...), where (k)

(k)

(k)

(d(k) ) = (δ (k) , d0 , d1 , ..., d )

(6.159)

is a type of a space which contains comitants that have focus pseudo-quantities of the differential system s(1, m1 , ..., m ) as coefficients. Existence of spaces of comitants (6.158) is argued by Lemma 2.4. These spaces are characterized by the fact that they contain at least one homogeneous polynomial with respect to x, y, in which the coefficients are generalized focus pseudo-quantities, which are characterized by the fact that on the invariant variety V from (5.11) their semi-invariants, except for a numerical constant, go to the corresponding focus quantity Lk . Thus, we obtain (0,0,...,0)

(0,1,...,0)

(δ (k) ,d

(k)

,d

(k)

,...,d

(k)

)

0 1  the sequence of spaces R = S1,m1 ,...,m , S1,m1 ,...,m ,..., S1,m1 ,...,m ,...   from S1,m1 ,...,m . Using them, we form a graded subalgebra S1,m1 ,...,m , which satisfies the inclusion S1 ,m1 ,...,m ⊂ S1,m1 ,...,m ,

where according to Proposition 4.1, it follows that between their Krull dimensions, the following inequality holds: (S1 ,m1 ,...,m ) ≤ (S1,m1 ,...,m ). From formula (2.54) and Remark 2.2, we obtain

   (S1,m1 ,...,m ) = 2 mi +  + 3.

(6.160)

(6.161)

i=1

Similar to the examples considered in Sections 6.1–6.6 and using (6.160) and (6.161), it can be shown that the following is true Lemma 6.1. Maximal number of algebraically independent generalized focus pseudo-quantities in the center and focus problem for differential system

146

The Center and Focus Problem

(6.156) does not exceed the number from (6.161), i.e. does not exceed maximal number of all possible nonzero coefficients of the right-hand sides of this system minus one. Considering that generalized focus pseudo-quantities, being semi-invariants in comitants that are contained in all homogeneous spaces of the algebra S1 ,m1 ,...,m of the system s(1, m1 , ..., m ) from (1.1)–(1.2) on the variety V from (5.9) or, what is the same, from (5.11), are being transformed up to a numerical factor in the focus quantities L1 , L2 ,..., Lk ,... of this system, using Lemma 6.1, we obtain that there takes place Theorem 6.10. Maximal number of algebraically independent focus quantities of differential system (6.156), that take part in solving the center and focus problem, does not exceed the number from (6.161), i.e., does not exceed maximal number of all possible nonzero coefficients of the right-hand sides of this system minus one. Recall that for the differential systems s(1, 2) and s(1, 3) the number of essential conditions of center is ω = 3 and 5, respectively, and for the differential system s(1, 2, 3), according to one hypothesis, it is ω ≤ 13. From Theorem 6.10, we have that maximal number of algebraically independent focus quantities of the differential system s(1, 2) does not exceed 9, for the differential system s(1, 3) it does not exceed 11, and for the differential system s(1, 2, 3) it does not exceed 17. These arguments and Proposition 4.2 with the variety V from (5.9), which is equivalent to (5.11), and the previously defined algebra S1 ,m1 ,...,m , allow us to conclude that the following can be true Hypothesis 6.3. Number of essential conditions of center ω (5.2), that solve the center and focus problem for differential system (6.156), which at the origin has a singular point of the second group, does not exceed the number from (6.161), i.e., does not exceed maximal number of all possible nonzero coefficients of the right-hand sides of this system minus one. Observation 6.1. Equality (6.161) shows that the quantity  is equal to maximal number of all possible nonzero coefficients of the right-hand sides of system (6.156) minus one. These results were first reported in the paper [38]. Note that in Section 3.3 from Sibirsky’s monograph [38], it is determined that an estimate of the lower limit of number of essential conditions of center when differential system (1.1) contains all nonhomogeneities of degree from 1 to q (q > 1). In other words, for the differential system s(Γ), where Γ = {1, 2, ..., q}, there takes place Theorem 6.11. [38]. Number of essential conditions of center is not less than q 2 − q for even q and than q 2 − q − 1 for odd q.

Algebraically Independent Focus Quantities

147

Adapting the result from Theorem 6.11 for the differential system s(1, 2, ..., q), we find 

q  (S1,2,...,q ) = 2 i + q − 1 + 3 = q 2 + 3q − 1. i=2

We obtain that between the estimates from Theorems 6.10 and 6.11 [38] there are relations (S1,2,...,q ) = q 2 + 3q − 1 = [q 2 − q] + 4q − 1 or [q 2 − q − 1] + 4q . ' () * ' () * q=2k

q=2k+1

So we get that for the differential system s(1, 2, ..., q) the estimation from Theorem 6.10 is greater than the estimation from Theorem 6.11 [38] by 4q − 1 if q is even and by 4q if q is odd.

6.8

Comments to Chapter Six

The results of this chapter allowed us to approach to an estimation for the upper bound of the number of algebraically independent focus quantities, which are involved in solving the center and focus problem for any differential system of the form (1.1)–(1.2), the problem formulated more than 130 years ago by the French mathematician Poincar´e. These results made it possible to propose a valid hypothesis on the upper bound of the number of essential focus quantities, which are involved in solving the considered problem for any differential system of the form (1.1)–(1.2). In these studies, an important role was played by Hilbert series for Sibirsky algebras of invariants and comitants and also by Lie algebra of representation of centro-affine group in the space of phase variables and coefficients of systems of the form (1.1)–(1.2). The main result is that the number of algebraically independent focus quantities for the differential system s(1, m1 , ..., m ) of the form (6.156), which has at the origin a singular point of center or focus type does not exceed the   number  = 2( mi + ) + 3, which is the Krull dimension of the Sibirsky i=1

graded algebra of comitants S1,m1 ,...,m of system (6.156). This is the solution of the generalized center and focus problem for the system s(1, m1 , ..., m ) of the form (6.156).

7 On the Upper Bound of the Number of Functionally Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for Lyapunov System

7.1

Lie Operators of Representation of the Rotation Group SO(2, R) in the Space of Coefficients of Lyapunov System (5.6)

Consider the center and focus problem for classical system of differential equations (5.6), which has the general form x˙ = y +

` X

Pmi (x, y), y˙ = −x +

i=1

where

` X

Qmi (x, y) (` < ∞),

(7.1)

i=1

 mi  X mi i 1 mi −k k ak x y , Pmi (x, y) = k k=0  mi  X mi i 2 mi −k k Qmi (x, y) = ak x y (mi ∈ Γ; i = 1, `). k

(7.2)

k=0

The set Γ = {mi }`i=1 is a finite set of different natural numbers. Recall that we will call system (7.1) as a Lyapunov system [20] and denote it by sL(1, m1 , ..., m` ). We study the action of the rotation group SO(2, R), given by formulas x = x cos ϕ + y sin ϕ, y = −x sin ϕ + y cos ϕ (0 ≤ ϕ < π),

(7.3)

on system (7.1)–(7.2). Due to transformations (7.3) in system (7.1)–(7.2), we obtain x˙ = y +

` X

Pmi (x, y), y˙ = −x +

i=1

DOI: 10.1201/9781003193074-8

` X

Qmi (x, y) (` < ∞),

(7.4)

i=1

149

150

The Center and Focus Problem i

where x, y have the form (7.3), and the coefficients bkj (i = 1, ; j = 1, 2; k = 1, mi ) in the polynomials mi   mi i 1 mi −k k y , Pmi (x, y) = bk x k k=0 (7.5) mi   mi i 2 mi −k k Qmi (x, y) = bk x y (mi ∈ Γ; i = 1, ) k k=0

will be written in the form i

i

b2k

i a2k

i

i

i

b1k = a1k cosmi +1 ϕ + [(mi − k)a1k+1 − ka1k−1 + a2k ] cosmi ϕ sin ϕ +o(sin ϕ), i

=

cos

mi +1

ϕ+

i [−a1k

i k)a2k+1

+ (mi − +o(sin ϕ).

−

i2 kak−1 ] cosmi

(7.6)

ϕ sin ϕ

Note that o(sin ϕ) is a linear function of coefficients of system (7.1)–(7.2) and contains sin ϕ of degree not less than two in each term. From (7.3) and (7.6), according to Sections 1.2 and 1.4, it follows that a linear representation of the group SO(2, R) is a one-parameter group depending on the parameter ϕ, whose value in ϕ = 0 corresponds to the identity transformation i

i

x = x, y = y, bkj = akj (i = 1, ; j = 1, 2; k = 1, mi ). With this, according to formula (1.35), we obtain ξ 1 (x, y) = y, ξ 2 = −x, i

i

i

i

η 1k = (mi − k)a1k+1 − ka1k−1 + a2k (i = 1, ; k = 0, mi ), i

i

i

i

2 η 2k = −a1k + (mi − k)a2k+1 − kak−1 (i = 1, ; k = 0, mi ).

Substituting these equalities in (1.38)–(1.39), we find that there takes place Theorem 7.1. Lie operator of representation of the group SO(2, R) in the space E N +2 (x, y, A) of system (7.1)–(7.2) has the form X=y where D=

mi   

∂ ∂ −x + D, ∂x ∂y

(7.7)

 i

i

i=1 k=0 i

∂

i

[(mi − k)a1k+1 − ka1k−1 + a2k ] i

i

+[−a1k + (mi − k)a2k+1 − ka2k−1 ]

∂ i

∂a2k

i

∂a1 + k .

(7.8)

On the Upper Bound of Focus Quantities

151

By A we denote a totality of coefficients of nonlinearities of the right parts of system (7.1)–(7.2). Consequence 7.1. Lie operator of representation of the group SO(2, R) in the space of coefficients E N (A) of system (7.1)–(7.2) has the form (7.8). Remark 7.1. Theorem 7.1 and Consequence 7.1 are known from the papers [33,34] for the system s(1, m1 , ..., m ) of general form. Remark 7.2. Using defining equations (1.66), it can be verified that operators (7.7)–(7.8) are admitted by the Lyapunov system sL(1, m1 , ..., m ) from (7.1)–(7.2).

7.2

Comitants of System (7.1) − (7.2) for the Rotation Group and Concept of Functional Basis

Definition 7.1. The polynomial F (x, y, A) in coefficients of system (7.1)– (7.2) and the phase variables x, y is called algebraic comitant of this system under the rotation group SO(2, R) from (7.3), if for all admissible A, x, y and ϕ the following identity holds F (x, y, B) = F (x, y, A),

(7.9)

where A (B) is a totality of coefficients of system (7.1)–(7.2) ((7.4)–(7.5)), and (x, y), (x, y) are the phase variables of the same systems. If the comitant F of system (7.1)–(7.2) does not depend on the phase variables x, y, then, according to [38], it is called invariant of the indicated system under the rotation group. Definition 7.2. A totality of algebraic comitants of system (7.1)–(7.2) with respect to the rotation group {Fα (x, y, A), α ∈ N+ }

(7.10)

is called a functional basis of comitants of indicated system under the group SO(2, R) if any comitant F (x, y, A) of considered system under the rotation group can be represented as unambiguous function of comitants (7.10). From the papers [33,34] it follows that there takes place Theorem 7.2. In order that the polynomial F (x, y, A) to be a comitant of system (7.1)–(7.2) under the rotation group (7.3), i.e., to satisfy equality (7.9), it is necessary and sufficient for it to satisfy the equation X(F ) = 0,

(7.11)

152

The Center and Focus Problem

where X is a Lie operator from (7.7)–(7.8). Using Theorem 7.2, it is easy to verify that there takes place Consequence 7.2. In order that the polynomial I(A) to be an invariant of system (7.1)–(7.2) under the rotation group (7.3), it is necessary and sufficient for it to satisfy the equation D(I) = 0,

(7.12)

where D is a Lie operator from (7.8). Remark 7.3. From (7.11) and (7.12), it follows that comitants and invariants of the Lyapunov system sL(1, m1 , ..., m ) (7.1)–(7.2) are solutions of homogeneous first-order partial differential equations with Lie operators (7.7) and (7.8). Note that the number of variables involved in operators (7.7) and (7.8) is equal, respectively, to

N =2

 

 mi +  + 1

(7.13)

i=1

and

N1 = 2

 

 mi +  .

(7.14)

i=1

From the general theory of equations of types (7.11) and (7.12) (see, e.g.,[27]) it is known that maximal number of functionally independent solutions is equal to N − 1 and N1 − 1, respectively. Example 7.1. Consider differential system (5.16) with quadratic nonlinearities written in a tensor notation. It is known from the paper [37] that generators of Sibirsky algebra for system (5.16) are the following invariants and comitants: α β α β γ pq α β γ pq I 1 = aα α , I2 = aβ aα , I3 = ap aαq aβγ ε , I4 = ap aβq aαγ ε , β γ pq α β γ δ pq I 5 = aα p aγq aαβ ε , I6 = ap aγ aαq aβδ ε , β γ δ pq rs α β γ δ pq rs I 7 = aα pr aαq aβs aγδ ε ε , I8 = apr aαq aδs aβγ ε ε , β γ δ pq rs α β γ δ ν pq I9 = aα pr aβq aγs aαδ ε ε , I10 = ap aδ aν aαq aβγ ε , μ pq rs β γ δ β γ δ aβs aαγ aδμ ε ε , I12 = apα aqr aβs aαδ aμγμ εpq εrs , I11 = apα aqr μ pq rs μ ν pq rs β γ δ α β γ δ I13 = aα p aqr aγs aαβ aδμ ε ε , I14 = ap ar aαq aβs aγδ aμν ε ε , β γ δ μ ν pq rs kl I15 = aα pr aqk aαs aδl aβγ aμν ε ε ε , μ ν β γ δ τ pq rs I16 = aα p ar aδ aαq aβs aγτ aμν ε ε .

(7.15)

On the Upper Bound of Focus Quantities

153

β p α q α β γ α β γ K1 = aα αβ x , K2 = aα x x εpq , K3 = aβ aαγ x , K4 = aγ aαβ x , β α γ δ aβγδ xγ xδ , K7 = aα K5 = apαβ xα xβ xq εpq , K6 = aαβ βγ aαδ x x , β γ δ α β γ δ pq α β γ δ pq K 8 = aα γ aδ aαβ x , K9 = aαp aγq aβδ x ε , K10 = aαp aδq aβγ x ε , γ β γ q α β δ μ K11 = apα aα βγ x x x εpq , K12 = aβ aαγ aδμ x x , γ β δ μ α β γ δ μ pq K13 = aα γ aαβ aδμ x x , K14 = ap aαq aβδ aγμ x ε ,

(7.16)

β γ δ μ pq α β γ δ μ pq K15 = aα p aαq aβμ aγδ x ε , K16 = ap aβq aαμ aγδ x ε , γ β δ μ ν α β γ δ μ ν pq K17 = aα βν aαγ aδμ x x x , K18 = aμp aαq aβν aγδ x x ε , μ ν pq μ ν pq rs β γ δ α β γ δ K19 = aα p aν aαq aβμ aγδ x ε , K20 = apr aνq aαs aβγ aδμ x ε ε .

We write system (5.16) in the form (7.1)–(7.2). According to notation (5.18), it will take the form x˙ = y + gx2 + 2hxy + ky 2 , y˙ = −x + x2 + 2mxy + ny 2 ,

(7.17)

for which Lie operators (7.7)–(7.8) will have the form

X=y

∂ ∂ −x + D, ∂x ∂y

(7.18)

where ∂ ∂ ∂ + (−g + k + m) + (2h + n) ∂g ∂h ∂k ∂ ∂ ∂ +(−g + 2m) + (−h −  + n) − (k + 2m) . ∂ ∂m ∂n D = (2h + )

(7.19)

Then, taking into account notation (5.18) for invariants and comitants (7.15)–(7.16), obtained for system (7.17), we have D(Ij ) = 0 (j = 1, 16), X(Ki ) = 0 (i = 1, 20).

(7.20)

Using equalities (7.20) and Theorem 7.2, it is proved the following Lemma 7.1. Centro-affine invariants and comitants (7.15)–(7.16) of system (5.16) are invariants and comitants of the rotation group SO(2, R) for system (7.17). The reciprocal of Lemma 7.1 is not always true.

154

The Center and Focus Problem

7.3

General Formulas that Interconnect Coefficients of Comitants of the Lyapunov System sL(1, m1 , ..., m ) Among Themselves with Respect to the Rotation Group

We write the comitant K of the Lyapunov system (7.1)–(7.2) with respect to the rotation group in the form K = A0 xm + A1 xm−1 y + A2 xm−2 y 2 + A3 xm−3 y 3 + A4 xm−4 y 4 +A5 xm−5 y 5 + A6 xm−6 y 6 + ... + Am−7 x7 y m−7 + Am−6 x6 y m−6 +Am−5 x5 y m−5 + Am−4 x4 y m−4 + Am−3 x3 y m−3 + Am−2 x2 y m−2

(7.21)

+Am−1 x1 y m−1 + Am y m , where Ai (i = 1, m) are polynomials in coefficients of system (7.1)–(7.2). We consider equality (7.11) taking into account the form of Lie operator (7.7). Using (7.21) and Lie operator (7.7), we obtain X(K) = y

∂K ∂K −x + D(K). ∂x ∂y

(7.22)

Then the terms on the right side (7.22), considering (7.21), are written in the form ∂K = mA0 xm−1 + (m − 1)A1 xm−2 y + (m − 2)A2 xm−3 y 2 ∂x +(m − 3)A3 xm−4 y 3 + (m − 4)A4 xm−5 y 4 + (m − 5)A5 xm−6 y 5 y

+(m − 6)A6 xm−7 y 6 + ... + 7Am−7 x6 y m−6 + 6Am−6 x5 y m−5 +5Am−5 x4 y m−4 + 4Am−4 x3 y m−3 + 3Am−3 x2 y m−2 +2Am−2 xy m−1 + Am−1 y m , ∂K = −A1 xm − 2A2 xm−1 y − 3A3 xm−2 y 2 − 4A4 xm−3 y 3 ∂y −5A5 xm−4 y 4 − 6A6 xm−5 y 5 − ... − (m − 7)Am−7 x8 y m−8

−x

−(m − 6)Am−6 x7 y m−7 − (m − 5)Am−5 x6 y m−6 −(m − 4)Am−4 x5 y m−5 − (m − 3)Am−3 x4 y m−4 −(m − 2)Am−2 x3 y m−3 − (m − 1)Am−1 x2 y m−2 − mAm xy m−1 , D(K) = D(A0 )xm + D(A1 )xm−1 y + D(A2 )xm−2 y 2 +D(A3 )xm−3 y 3 + D(A4 )xm−4 y 4 + D(A5 )xm−5 y 5 +D(A6 )xm−6 y 6 + ... + D(Am−7 )x7 y m−7 + D(Am−6 )x6 y m−6 +D(Am−5 )x5 y m−5 + D(Am−4 )x4 y m−4 + D(Am−3 )x3 y m−3 +D(Am−2 )x2 y m−2 + D(Am−1 )x1 y m−1 + D(Am )y m .

(7.23)

On the Upper Bound of Focus Quantities

155

Considering relation (7.11) and the last expressions from X(K) = 0, we have the following equalities: xm : D(A0 ) − A1 = 0, xm−1 y : mA0 + 2A2 + D(A1 ) = 0, xm−2 y 2 : (m − 1)A1 − 3A3 + D(A2 ) = 0, xm−3 y 3 : (m − 2)A2 − 4A4 + D(A3 ) = 0, xm−4 y 4 : (m − 3)A3 − 5A5 + D(A4 ) = 0, xm−5 y 5 : (m − 4)A4 − 6A6 + D(A5 ) = 0, xm−6 y 6 : (m − 5)A5 − 7A7 + D(A6 ) = 0,

xm−7 y 7 : (m − 6)A6 − 8A8 + D(A7 ) = 0, .............................................................. x6 y m−6 : 7Am−7 − (m − 5)Am−5 + D(Am−6 ) = 0, x5 y m−5 : 6Am−6 − (m − 4)Am−4 + D(Am−5 ) = 0, x4 y m−4 : 5Am−5 − (m − 3)Am−3 + D(Am−4 ) = 0, x3 y m−3 : 4Am−4 − (m − 2)Am−2 + D(Am−3 ) = 0, x2 y m−2 : 3Am−3 − (m − 1)Am−1 + D(Am−2 ) = 0, xy m−1 : 2Am−2 − (m)Am + D(Am−1 ) = 0, y m : Am−1 + D(Am ) = 0. From this and Theorem 7.2, we obtain that there holds Theorem 7.3. Polynomial (7.21) is a comitant of the Lyapunov system sL(1, m1 , ..., m ) from (7.1) with respect to the rotation group SO(2, R) if and only if its coefficients satisfy the equalities A1 = D(A0 ), D(Am ) = −Am−1 , 1 Ak = [D(Ak−1 ) + (m − k + 2)Ak−2 ] (k = 2, m), k

(7.24)

where Ai (i = 0, m) are taken from (7.21), and D is from (7.8). For example, we can verify the validity of Theorem 7.3 on comitants (7.16) with notation (5.18) for system (7.17) with operators (7.19). Consequence 7.3. If the coefficient A0 at the highest degree of x of the comitant K from (7.21) of the Lyapunov system (7.1)–(7.2) with respect to the rotation group SO(2, R) and its degree with respect to x and y are known, then the remaining coefficients can be constructed by formulas (7.24). By analogy with the comitants of the group of centro-affine transformations for system (1.1)–(1.2), we will call the coefficient A0 of the comitant K

156

The Center and Focus Problem

from (7.21) of the Lyapunov system (7.1)–(7.2) with respect to the rotation group SO(2, R) as a semi-invariant. It can be easily verified that there takes place Observation 7.1. If the comitants K1 , K2 , ..., Kr of the Lyapunov system (7.1)–(7.2) with respect to the group SO(2, R) are functionally independent, then its semi-invariants can also be functionally independent. There from it follows Remark 7.4. The number of functionally independent semi-invariants for the functionally independent comitants K1 , K2 , ..., Kr of the Lyapunov system (7.1)–(7.2) with respect to the group SO(2, R) does not exceed r.

7.4

On the Invariance of Focus Quantities in the Center and Focus Problem with Respect to the Rotation Group

In the paper [38, p. 84], it is shown that there takes place Remark 7.5. The conditions of existence of center for the Lyapunov system sL(1, m1 , ..., m ) from (7.1)–(7.2) are invariants of this system with respect to the rotation group SO(2, R). It is known that focus quantities for the Lyapunov system sL(1, m1 , ..., m ) from (5.6) are ambiguously constructed since equations (5.5) with the Lyapunov function (5.4) contain arbitrary constants, which can take different values. We will show on some examples that focus quantities in the center and focus problem for the Lyapunov system sL(1, m1 , ..., m ) from (7.1)–(7.2) can be invariants and semi-invariants of this system with respect to the rotation group SO(2, R). For this, we use the results obtained in Section 7.3. Consider the Lyapunov system sL(1, 2) from (7.17) and Lie operator (7.18)–(7.19), admitted by this system. In the paper of A. P. Sadovsky [35, p. 110], three focus quantities are given that solve the center and focus problem for the system sL(1, 2) from (7.17) and have the following form: L1 S =

1 (−gh − hk + gl + lm − kn + mn), 2

1 (62g 3 h − 2gh3 + 119g 2 hk − 2h3 k + 62ghk 2 + 5hk 3 24 −62g 3 l + 27gh2 l − 63g 2 kl + 29h2 kl − 15gk 2 l − 8ghl2 + 15hkl2 L2 S =

−5gl3 + 101g 2 hm + 138ghkm + 37hk 2 m − 175g 2 lm − 6h2 lm

On the Upper Bound of Focus Quantities −116gklm − 15k 2 lm − 13hl2 m − 5l3 m + 54ghm2 + 54hkm2 −159glm2 − 53klm2 − 46lm3 + 6g 3 n + 37gh2 n + 72g 2 kn + 39h2 kn +57gk 2 n + 5k 3 n + 14ghln + 68hkln − 33gl2 n + 15kl2 n − 72g 2 mn −6h2 mn + 34gkmn + 32k 2 mn − 42hlmn − 38l2 mn − 109gm2 n −3km2 n − 46m3 n + 48ghn2 + 79hkn2 − 48gln2 + 39kln2 −29hmn2 − 71lmn2 − 6gn3 + 38kn3 − 38mn3 ), 1 (−44725g 5 h + 11142g 3 h3 − 88gh5 − 124537g 4 hk 2304 +30842g 2 h3 k − 88h5 k − 121728g 3 hk 2 + 27186gh3 k 2 − 45492g 2 hk 3 L3 S =

+7486h3 k 3 − 2651ghk 4 + 925hk 5 + 44725g 5 l − 51066g 3 h2 l −1216gh4 l + 84372g 4 kl − 95044g 2 h2 kl − 1704h4 kl + 53320g 3 k 2 l −44602gh2 k 2 l + 10096g 2 k 3 l − 2880h2 k 3 l − 465gk 4 l + 28368g 3 hl2 −3362gh3 l2 + 7708g 2 hkl2 − 5802h3 kl2 − 13582ghk 2 l2 − 3650hk 3 l2 +2436g 3 l3 + 1858gh2 l3 + 9332g 2 kl3 − 1700h2 kl3 + 3650gk 2 l3 −875ghl4 + 465hkl4 − 925gl5 − 157320g 4 hm + 7528g 2 h3 m −356424g 3 hkm + 19904gh3 km − 264096g 2 hk 2 m + 12376h3 k 2 m −67288ghk 3 m − 2296hk 4 m + 211165g 4 lm − 72038g 2 h2 lm +888h4 lm + 310816g 3 klm − 109100gh2 klm + 144112g 2 k 2 lm −26630h2 k 2 lm + 17812gk 3 lm − 465k 4 lm + 67344g 2 hl2 m +2776h3 l2 m + 12024ghkl2 m − 7552hk 2 l2 m + 14808g 2 l3 m +978h2 l3 m + 17844gkl3 m + 3650k 2 l3 m − 1800hl4 m − 925l5 m −214106g 3 hm2 − 4560gh3 m2 − 214106g 2 hkm2 − 4560h3 km2 −29106ghk 2 m2 − 29106hk 3 m2 + 389610g 3 lm2 − 16864gh2 lm2 +249820g 2 klm2 − 32896h2 klm2 + 131634gk 2 lm2 + 7716k 3 lm2 +60798ghl2 m2 + 9126hkl2 m2 + 15478gl3 m2 + 8512kl3 m2 −131528g 2 hm3 − 188448ghkm3 − 56920hk 2 m3 + 350410g 2 lm3 +7632h2 lm3 + 83832gklm3 + 40842k 2 lm3 + 15912hl2 m3 +3106l3 m3 − 29432ghm4 − 29432hkm4 + 152800glm4 +60456klm4 + 25560lm5 − 4560g 5 n − 49528g 3 h2 n −2168gh4 n − 56129g 4 kn − 71382g 2 h2 kn − 2656h4 kn −86332g 3 k 2 n − 8356gh2 k 2 n − 42376g 2 k 3 n + 11242h2 k 3 n −3576gk 4 n + 925k 5 n − 8288g 3 hln − 21396gh3 ln −107608g 2 hkln − 28148h3 kln − 78028ghk 2 ln − 6580hk 3 ln

157

158

The Center and Focus Problem +41832g 3 l2 n − 3496gh2 l2 n + 29628g 2 kl2 n − 28690h2 kl2 n −7552gk 2 l2 n − 3650k 3 l2 n + 7132ghl3 n − 5780hkl3 n −520gl4 n + 465kl4 n + 44753g 4 mn − 87182g 2 h2 mn +888h4 mn − 78864g 3 kmn − 108972gh2 kmn − 155168g 2 k 2 mn −11358h2 k 2 mn − 60956gk 3 mn − 3221k 4 mn + 51096g 2 hlmn +9632h3 lmn − 124544ghklmn − 50104hk 2 lmn + 144264g 2 l2 mn +16334h2 l2 mn + 63884gkl2 mn − 1522k 2 l2 mn + 5272hl3 mn −1445l4 mn + 173116g 3 m2 n − 43264gh2 m2 n + 17830g 2 km2 n −59296h2 km2 n + 82532gk 2 m2 n − 25890k 3 m2 n + 121900ghlm2 n −26836hklm2 n + 146916gl2 m2 n + 39066kl2 m2 n + 222962g 2 m3 n +7632h2 m3 n − 103272gkm3 n − 18814k 2 m3 n + 53184hlm3 n +38574l2 m3 n + 125304gm4 n + 32960km4 n + 25560m5 n −56942g 3 hn2 − 23378gh3 n2 − 141270g 2 hkn2 − 27690h3 kn2 −60328ghk 2 n2 + 6856hk 3 n2 + 48734g 3 ln2 − 38598gh2 ln2 −11808g 2 kln2 − 81368h2 kln2 − 38236gk 2 ln2 − 3700k 3 ln2 +22272ghl2 n2 − 43344hkl2 n2 + 12584gl3 n2 − 4080kl3 n2 −57664g 2 hmn2 + 6856h3 mn2 − 184664ghkmn2 −49232hk 2 mn2 + 202966g 2 lmn2 + 36790h2 lmn2 +13848gklmn2 − 28284k 2 lmn2 + 43488hl2 mn2 +11604l3 mn2 + 28654ghm2 n2 − 68410hkm2 n2 +243030glm2 n2 + 34596klm2 n2 + 37272hm3 n2 + 75942lm3 n2 −288g 3 n3 − 43772gh2 n3 − 48542g 2 kn3 − 64906h2 kn3 −30696gk 2 n3 + 3100k 3 n3 + 4176ghln3 − 85408hkln3 +31676gl2 n3 − 20456kl2 n3 + 55598g 2 mn3 + 21434h2 mn3 −58400gkmn3 − 31408k 2 mn3 + 69384hlmn3 + 39200l2 mn3 +100760gm2 n3 − 6790km2 n3 + 40474m3 n3 − 18481ghn4 −56701hkn4 + 25249gln4 − 30484kln4 + 32968hmn4 +43993lmn4 + 3408gn5 − 16917kn5 + 16917mn5 ).

In Li S, the letter S emphasizes that Li S are focus quantities of A. P. Sadovsky. Remark 7.6. Note that for focus quantities (7.25) using Lie operator (7.19), we find D(L1 S) = 0, D(L2 S) = 0, D(L3 S) = 0, whence it follows that L1 S is an invariant of the system sL(1, 2) under the rotation group SO(2, R), and L2 S, L3 S are not.

On the Upper Bound of Focus Quantities

159

In view of Theorems 7.2 and 7.3 and operators (7.18)–(7.19), we find that to the focus quantities L2 S and L3 S from (7.25), the following comitants of the rotation group correspond: K(L2 S) = L2 Sx4 + A1 x3 y + A2 x2 y 2 + A3 xy 3 + A4 y 4 , where

1 [D(A1 ) + 4L2 S], 2 1 1 A3 = [D(A3 ) + 2A2 ], A4 = [D(A3 ) + 2A2 ], 3 4

(7.25)

A1 = D(L2 S), A2 =

(7.26)

and K(L3 S) = L3 Sx12 + A1 x11 y + A2 x10 y 2 + A3 x9 y 3 + A4 x8 y 4 +A5 x7 y 5 + A6 x6 y 6 + A7 x5 y 7 + A8 x4 y 8 + A9 x3 y 9 + A10 x2 y 10 +A11 xy

11

(7.27)

12

+ A12 y ,

for 1 1 [D(A1 ) + 12L3 S], A3 = [D(A2 ) + 11A1 ], 2 3 1 1 A4 = [D(A3 ) + 10A2 ], A5 = [D(A4 ) + 9A3 ], 4 5 1 1 A6 = [D(A5 ) + 8A4 ], A7 = [D(A6 ) + 7A5 ], 6 7 1 1 A8 = [D(A7 ) + 6A6 ], A9 = [D(A8 ) + 5A7 ], 8 9 1 1 A10 = [D(A9 ) + 4A8 ], A11 = [D(A10 ) + 3A9 ], 10 11 1 A12 = [D(A11 ) + 2A10 ]. 12

A1 = D(L3 S), A2 =

(7.28)

Using Lie operator (7.18)–(7.19) from (7.26)–(7.27) and (7.28)–(7.29) we find X[K(L2 S)] = X[K(L3 S)] = 0.

(7.29)

Therefore, from Remark 7.6 and equalities (7.30), we obtain that the following holds: Lemma 7.2. The focus quantity L1 S is an invariant, but L2 S and L3 S are semi-invariants of the system sL(1, 2) from (7.17) with respect to the rotation group SO(2, R). Remark 7.7. The other three focus quantities that solve the center and focus problem for the system sL(1, 2) from (7.17) were introduced to us by Professor Iu. F. Calin, built by another method. They have the form L1 C = L1 S =

1 (−gh − hk + gl + lm − kn + mn), 2

160

The Center and Focus Problem 1 (53g 3 h + 16gh3 + 95g 2 hk + 16h3 k + 47ghk 2 + 5hk 3 24 −53g 3 l + 18gh2 l − 48g 2 kl + 38h2 kl − 15gk 2 l − 17ghl2 + 15hkl2 L2 C =

−5gl3 + 62g 2 hm + 84ghkm + 22hk 2 m − 127g 2 lm − 24h2 lm − 86gklm −15k 2 lm − 22hl2 m − 5l3 m + 24ghm2 + 24hkm2 − 90glm2 − 38klm2 −16lm3 + 6g 3 n + 70gh2 n + 63g 2 kn + 90h2 kn + 42gk 2 n + 5k 3 n −10ghln + 86hkln − 42gl2 n + 15kl2 n − 63g 2 mn − 24h2 mn + 10gkmn +17k 2 mn − 84hlmn − 47l2 mn − 70gm2 n − 18km2 n − 16m3 n + 63ghn2 +127hkn2 − 63gln2 + 48kln2 − 62hmn2 − 95lmn2 − 6gn3 +53kn3 − 53mn3 ), 1 (−31393g 5 h − 17022g 3 h3 − 1144gh5 − 77985g 4 hk 2304 −27330g 2 h3 k − 1144h5 k − 67264g 3 hk 2 − 8666gh3 k 2 − 21292g 2 hk 3 L3 C =

+1642h3 k 3 + 305ghk 4 + 925hk 5 + 31393g 5 l − 27138g 3 h2 l − 10960gh4 l +50720g 4 kl − 70216g 2 h2 kl − 13080h4 kl + 30636g 3 k 2 l − 41898gh2 k 2 l +5940g 2 k 3 l − 5844h2 k 3 l − 465gk 4 l + 35076g 3 hl2 − 1706gh3 l2 +16672g 2 hkl2 − 14946h3 kl2 − 6030ghk 2 l2 − 3650hk 3 l2 + 828g 3 l3 +7270gh2 l3 + 6664g 2 kl3 − 3560h2 kl3 + 3650gk 2 l3 + 265ghl4 + 465hkl4 −925gl5 − 79756g 4 hm − 30792g 2 h3 m − 164120g 3 hkm − 37792gh3 km −106536g 2 hk 2 m − 7000h3 k 2 m − 21512ghk 3 m + 660hk 4 m + 119405g 4 lm +4582g 2 h2 lm + 3096h4 lm + 157960g 3 klm − 48148gh2 klm +72848g 2 k 2 lm − 19738h2 k 2 lm + 9500gk 3 lm − 465k 4 lm + 74424g 2 hl2 m +12424h3 l2 m + 31600ghkl2 m + 6844g 2 l3 m + 7530h2 l3 m + 12508gkl3 m +3650k 2 l3 m − 660hl4 m − 925l5 m − 74114g 3 hm2 − 15376gh3 m2 −127550g 2 hkm2 − 15376h3 km2 − 60966ghk 2 m2 − 7530hk 3 m2 +168022g 3 lm2 + 41568gh2 lm2 + 177140g 2 klm2 + 57158gk 2 lm2 +3560k 3 lm2 + 52258ghl2 m2 + 19738hkl2 m2 + 4374gl3 m2 + 5844kl3 m2 = 29432g 2 hm3 − 41856ghkm3 − 12424hk 2 m3 + 104770g 2 lm3 +15376h2 lm3 + 82980gklm3 + 14946k 2 lm3 + 7000hl2 m3 − 1642l3 m3 −3096ghm4 − 3096hkm4 + 25904glm4 + 13080klm4 + 1144lm5 −4128g 5 n − 89564g 3 h2 n − 23784gh4 n − 41357g 4 kn − 183390g 2 h2 kn −25904h4 kn − 51912g 3 k 2 n − 91176gh2 k 2 n − 21132g 2 k 3 n − 4374h2 k 3 n −620gk 4 n + 925k 5 n + 30248g 3 hln − 43620gh3 ln − 64504g 2 hkln −82980h3 kln − 67900ghk 2 ln − 12508hk 3 ln + 43332g 3 l2 n + 17324gh2 l2 n +30812g 2 kl2 n − 57158h2 kl2 n − 3650k 3 l2 n + 21092ghl3 n − 9500hkl3 n

On the Upper Bound of Focus Quantities

161

+620gl4 n + 465kl4 n + 32573g 4 mn − 89970g 2 h2 mn + 3096h4 mn −32600g 3 kmn − 175220gh2 kmn − 61672g 2 k 2 mn − 52258h2 k 2 mn −21092gk 3 mn − 265k 4 mn + 162352g 2 hlmn + 41856h3 lmn −31600hk 2 lmn + 119932g 2 l2 mn + 60966h2 l2 mn + 67900gkl2 mn +6030k 2 l2 mn + 21512hl3 mn − 305l4 mn + 93296g 3 m2 n + 38762g 2 km2 n −41568h2 km2 n − 17324gk 2 m2 n − 7270k 3 m2 n + 175220ghlm2 n +48148hklm2 n + 91176gl2 m2 n + 41898kl2 m2 n + 78650g 2 m3 n +15376h2 m3 n + 43620gkm3 n + 1706k 2 m3 n + 37792hlm3 n +8666l2 m3 n + 23784gm4 n + 10960km4 n + 1144m5 n −68550g 3 hn2 − 78650gh3 n2 − 204974g 2 hkn2 − 104770h3 kn2 −119932ghk 2 n2 − 6844hk 3 n2 + 59766g 3 ln2 − 38762gh2 ln2 −177140h2 kln2 − 30812gk 2 ln2 − 6664k 3 ln2 + 61672ghl2 n2 −72848hkl2 n2 + 21132gl3 n2 − 5940kl3 n2 + 29432h3 mn2 −162352ghkmn2 − 74424hk 2 mn2 + 204974g 2 lmn2 +127550h2 lmn2 + 64504gklmn2 − 16672k 2 lmn2 +106536hl2 mn2 + 21292l3 mn2 + 89970ghm2 n2 −4582hkm2 n2 + 183390glm2 n2 + 70216klm2 n2 +30792hm3 n2 + 27330lm3 n2 − 93296gh2 n3 −59766g 2 kn3 − 168022h2 kn3 − 43332gk 2 n3 − 828k 3 n3 +32600ghln3 − 157960hkln3 + 51912gl2 n3 − 30636kl2 n3 +68550g 2 mn3 + 74114h2 mn3 − 30248gkmn3 − 35076k 2 mn3 +164120hlmn3 + 67264l2 mn3 + 89564gm2 n3 + 27138km2 n3 +17022m3 n3 − 32573ghn4 − 119405hkn4 + 41357gln4 −50720kln4 + 79756hmn4 + 77985lmn4 + 4128gn5

(7.30)

−31393kn + 31393mn ). 5

5

In Li C, the letter C emphasizes that Li C are focus quantities of Iu. F. Calin. Using the operator D from (7.19) and (7.31), we find D(L1 C) = D(L2 C) = D(L3 C) = 0. Therefore, from Remark 7.3, we have that focus quantities (7.31) are invariants of the system sL(1, 2) from (7.17) with respect to the rotation group SO(2, R). In a similar way that was used for focus quantities of the system sL(1, 2) one can find sequences of focus quantities for the systems sL(1, 3), sL(1, 4), sL(1, 2, 3) etc., which are invariants and semi-invariants of these systems relative to the rotation group SO(2, R).

162

The Center and Focus Problem

7.5

On the Upper Bound of the Number of Algebraically Independent Focus Quantities that Take Part in Solving the Center and Focus Problem for the Lyapunov System sL(1, m1 , ..., m )

Consider the set of comitants {F (x, y, A)} of the Lyapunov system sL(1, m1 , ..., m ) of type (7.9)–(7.10) with respect to the rotation group SO(2, R). It follows from Theorem 7.2 that equality (7.11) holds for them, where X is a Lie operator of linear representation of the rotation group SO(2, R) in the space E N +2 (x, y, A) of the system sL(1, m1 , ..., m ). Then, according to Remark 7.3, we have that maximal number of functionally independent solutions of equation (7.11) is equal to

   (7.31) μ=2 mi +  + 1, i=1

that corresponds to the number of coefficients in nonlinear part of the system sL(1, m1 , ..., m ) plus one. Since the focal quantities in the center and focus problem for the Lyapunov system sL(1, m1 , ..., m ) from (7.9)–(7.10), in general, are semi-invariants of this system with respect to the rotation group SO(2, R), then according to Remark 7.4, the number θ of functionally independent focus quantities for this system does not exceed the number μ from (7.32). Thus, we obtain Theorem 7.4. The number θ of functionally independent focus quantities in the center and focus problem for the Lyapunov system sL(1, m1 , ..., m ) does not exceed the number μ from (7.32), i.e., the following inequality holds 

  (7.32) θ≤μ=2 mi +  + 1. i=1

Hence, we have Remark 7.8. The number θ of functionally independent focus quantities from (44.2) in the center and focus problem for the Lyapunov system sL(1, m1 , ..., m ) from (7.9)–(7.10) does not exceed maximal number of all possible nonzero coefficients of this system minus one. Since the number of focus quantities that can solve the center and focus problem for the system sL(1, m1 , ..., m ) could be considered functionally independent, then it makes sense to assume that the following is true: Hypothesis 7.1. The number of essential focus quantities that solve the center and focus problem for the Lyapunov system sL(1, m1 , ..., m ) from (7.9)–(7.10) does not exceed maximal number of all possible nonzero coefficients of this system minus one.

On the Upper Bound of Focus Quantities

7.6

163

Comments to Chapter Seven

As it was mentioned earlier in Section 5.2, the system s(1, m1 , ..., m ) is written in the form x˙ = cx + dy +

 

Pmi (x, y), y˙ = ex + f y +

i=1

 

Qmi (x, y),

(7.33)

i=1

and the Lyapunov system sL(1, m1 , ..., m ) has the form x˙ = y +

 

Pmi (x, y), y˙ = −x +

i=1

 

Qmi (x, y),

(7.34)

i=1

where Pmi and Qmi are homogeneous polynomials of degree mi with respect to phase variables x and y. The set {1, m1 , ..., m } consists of a finite number of distinct positive integers. In Hypothesis 6.3, it was noted that the number of the essential center conditions that solve the center and focus problem for system (7.34) with a singular point of center or focus type at the origin does not exceed the number

   (7.35) =2 mi +  + 3, i=1

but Hypothesis 7.1 claims that the same number of the essential center conditions for system (7.35) is not greater than the number

μ=2

 

 mi + 

+ 1.

(7.36)

i=1

Since the system s(1, m1 , ..., m ) from (7.34), in the case of a singular point of center or focus type at the origin, can be reduced to the Lyapunov form sL(1, m1 , ..., m ) from (7.35) by a centro-affine transformation and time rescaling, then (7.36) implies that the number of the essential center conditions for system s(1, m1 , ..., m ) and for system sL(1, m1 , ..., m ) cannot exceed μ. Hence, we can claim the following Main Hypothesis 7.1. The number of essential focus quantities ω that solves the center and focus problem for system (7.34) with a singular point of center or focus type at the origin does not exceed the number μ from (7.36), i.e. does not exceed maximal number of all possible nonzero coefficients of this system minus one. This means an improvement by two units of the upper bound of the number of essential focus quantities ω, which are involved in solving the center and focus problem for system (7.33).

164

The Center and Focus Problem

The main result of this book can be formulated concisely as follows: Let   N = 2 (mi + 1) be the maximal possible number of nonzero coefficients of i=0

system (7.33), where m0 = 1. Then the number of algebraically independent focus quantities from (5.1) does not exceed N −1, which is the Krull dimension of Sibirsky algebra of comitants for system (7.33). It is also shown that this number could be reduced to N − 3, which is the Krull dimension of Sibirsky algebra of invariants for the mentioned system. It is assumed that the number of essential focus quantities ω from (5.2) does not exceed N − 1 and it can be improved up to N − 3, and their construction will begin with the first algebraically independent nonzero focus quantities obtained consecutively up to the mentioned estimations.

Appendix 1

The Polynomials Rk (b, e), that Define N1,4 (u, b, e) R0 (b, e) = 1 − e2 + 4e4 + e6 + 18e8 + 11e10 + 35e12 + 13e14 + 35e16 +11e18 + 18e20 + e22 + 4e24 − e26 + e28 + b(e2 + 5e4 + 13e6 + 26e8 +29e10 + 40e12 + 19e14 + 36e16 − 5e18 + 6e20 − 15e22 + e24 − 5e26 +2e28 − 2e30 ) + b2 (e2 + 8e4 + 16e6 + 26e8 + 27e10 + 20e12 + 12e14 −11e16 − 29e18 − 31e20 − 22e22 − 11e24 − 4e26 − 2e28 − e30 + e32 ) +b3 (e2 + 10e4 + 10e6 + 24e8 − 5e10 + 7e12 − 64e14 − 49e16 − 107e18 −55e20 − 58e22 − 10e24 − 10e26 + 3e28 + e30 ) + b4 (e2 + 6e4 + 9e6 +10e8 − 7e10 − 29e12 − 87e14 − 75e16 − 117e18 − 29e20 − 30e22 +26e24 + 2e26 + 17e28 − 3e30 + 4e32 ) + b5 (5e4 + 3e6 + 10e8 − 21e10 −38e12 − 82e14 − 76e16 − 72e18 + e20 + 20e22 + 44e24 + 32e26 + 17e28 +6e30 + 2e32 − 2e34 ) + b6 (2e4 + 3e6 − 2e8 − 29e10 − 36e12 − 84e14 −41e16 − 48e18 + 48e20 + 41e22 + 84e24 + 36e26 + 29e28 + 2e30 − 3e32 −2e34 ) + b7 (2e4 − 2e6 − 6e8 − 17e10 − 32e12 − 44e14 − 20e16 − e18 +72e20 + 76e22 + 82e24 + 38e26 + 21e28 − 10e30 − 3e32 − 5e34 ) +b8 (−4e6 + 3e8 − 17e10 − 2e12 − 26e14 + 30e16 + 29e18 + 117e20 + 75e22 +87e24 + 29e26 + 7e28 − 10e30 − 9e32 − 6e34 − e36 ) + b9 (−e8 − 3e10 +10e12 + 10e14 + 58e16 + 55e18 + 107e20 + 49e22 + 64e24 − 7e26 +5e28 − 24e30 − 10e32 − 10e34 − e36 ) + b10 (−e6 + e8 + 2e10 + 4e12 +11e14 + 22e16 + 31e18 + 29e20 + 11e22 − 12e24 − 20e26 − 27e28 − 26e30 −16e32 − 8e34 − e36 ) + b11 (2e8 − 2e10 + 5e12 − e14 + 15e16 − 6e18 + 5e20 −36e22 − 19e24 − 40e26 − 29e28 − 26e30 − 13e32 − 5e34 − e36 ) + b12 (−e10 +e12 − 4e14 − e16 − 18e18 − 11e20 − 35e22 − 13e24 − 35e26 − 11e28 −18e30 − e32 − 4e34 + e36 − e38 ), R1 (b, e) = −2e + 4e3 + e5 + 15e7 − 8e9 + 21e11 − 18e13 + 26e15 − 27e17 +6e19 − 19e21 + 7e23 − 6e25 + 2e27 − 2e29 + b(e + 5e3 + 8e5 + 5e7 + e9 +15e11 − 8e13 + 16e15 − 47e17 + 11e19 − 19e21 + 18e23 − 13e25 + 7e27 −4e29 + 4e31 ) + b2 (2e + 6e3 + 2e5 + 6e7 + 9e9 + 7e11 − 7e13 − 34e15 −24e17 + 2e19 + 11e21 + 6e23 + 5e25 + 5e27 + 4e29 + 2e31 − 2e33 ) + b3 (e +4e3 + 16e7 − 11e9 + 14e11 − 77e13 − e15 − 63e17 + 59e19 − 6e21 +52e23 − 3e25 + 19e27 − 3e29 − e31 ) + b4 (4e3 + 5e5 + 3e7 − 9e9 −26e11 − 60e13 + 4e15 − 35e17 + 88e19 − 11e21 + 51e23 − 24e25 +30e27 − 20e29 + 8e31 − 8e33 ) + b5 (4e3 + 7e7 − 23e9 − 19e11 − 49e13 +e15 + e17 + 73e19 + 12e21 + 30e23 − e25 − 9e27 − 16e29 − 10e31 −5e33 + 4e35 ) + b6 (e3 + e5 − 3e7 − 16e9 − 10e11 − 51e13 + 26e15 −11e17 + 96e19 − 3e21 + 60e23 − 45e25 − 4e27 − 38e29 − 7e31 + e33 165

166

Appendix 1

+3e35 ) + b7 (−3e5 + 2e7 − 6e9 − 21e11 − 23e13 + 18e15 + 26e17 + 73e19 +7e21 + 11e23 − 39e25 − 15e27 − 39e29 + 7e31 − 4e33 + 6e35 ) +b8 (6e7 − 20e9 − 24e13 + 61e15 + 9e17 + 88e19 − 49e21 + 20e23 −56e25 − 18e27 − 25e29 − e31 + e33 + 6e35 + 2e37 ) + b9 (−e7 − e9 +7e11 + 3e13 + 44e15 + 45e19 − 53e21 + 31e23 − 65e25 + 10e27 − 47e29 +12e31 + 14e35 + e37 ) + b10 (2e7 − 2e9 − e11 + 9e13 + 16e15 + 7e17 −6e19 − 12e21 − 12e23 − 9e25 − 21e27 − 7e29 + 14e31 + 14e33 + 8e35 ) +b11 (−4e9 + 7e11 + e13 + 14e15 − 23e17 + 11e19 − 35e21 + 18e23 −34e25 + 7e27 + 5e29 + 21e31 + 8e33 + 3e35 + e37 ) + b12 (2e11 − 4e13 −e15 − 15e17 + 8e19 − 21e21 + 18e23 − 26e25 + 27e27 − 6e29 + 19e31 −7e33 + 6e35 − 2e37 + 2e39 ), R2 (b, e) = 4e2 − 2e4 + 7e6 − 5e8 + 27e10 − 13e12 + 20e14 − 29e16 +14e18 − 17e20 + 5e22 − 14e24 + 3e26 − e28 + e30 + b(2e2 − 3e4 + 7e6 +e8 + 11e10 − 53e12 − 14e14 − 73e16 + 9e18 − 45e20 + 8e22 − 17e24 +21e26 − 5e28 + 2e30 − 2e32 ) + b2 (7e6 − 12e8 − 39e10 − 64e12 − 39e14 −37e16 + 2e18 − 16e20 + 16e22 + 17e24 + 16e26 − 3e28 + e30 − e32 + e34 ) +b3 (2e2 − e4 + e6 − 42e8 − 30e10 − 85e12 + 8e14 − 44e16 + 67e18 + e20 +81e22 + 16e24 + 32e26 − 6e28 + e30 − e32 ) + b4 (2e2 − 5e4 − 6e6 − 30e8 −18e10 − 53e12 + 65e14 − 3e16 + 159e18 + 44e20 + 143e22 + 9e24 + 40e26 −47e28 + 5e30 − 7e32 + 4e34 ) + b5 (e2 − 5e4 − 28e8 − 11e10 − 14e12 +84e14 + 57e16 + 172e18 + 51e20 + 89e22 − 23e24 − 25e26 − 43e28 + e30 −4e32 + 2e34 − 2e36 ) + b6 (−e4 − e6 − 26e8 + e10 − 12e12 + 103e14 + 67e16 +162e18 + 6e20 + 51e22 − 98e24 − 37e26 − 59e28 − 4e30 − 4e32 + 3e34 ) +b7 (−e6 − 13e8 + 7e10 + 22e12 + 103e14 + 51e16 + 104e18 − 20e20 − 9e22 −119e24 − 60e26 − 79e28 + 8e30 − 5e32 + 10e34 + e36 ) + b8 (−2e4 +2e6 − 11e8 + 26e10 + 23e12 + 87e14 + 79e18 − 92e20 − 24e22 −141e24 − 68e26 − 61e28 + 12e30 + 3e32 + 14e34 + 3e36 − e38 ) +b9 (3e6 − 4e8 + 22e10 + 5e12 + 55e14 − 30e16 + 30e18 − 137e20 −41e22 − 183e24 − 39e26 − 53e28 + 43e30 + 10e32 + 18e34 − 2e36 +e38 ) + b10 (−e8 + 7e10 − 2e12 − 4e14 − 53e16 − 42e18 − 104e20 −79e22 − 99e24 − 10e26 + 11e28 + 45e30 + 12e32 + 10e34 + 4e36 +3e38 ) + b11 (e6 − e8 + 5e10 − 9e12 − 6e14 − 36e16 − 10e18 − 67e20 −7e22 − 19e24 + 46e26 + 31e28 + 35e30 + 13e32 + 17e34 + 7e36 ) +b12 (−2e8 + 2e10 − 9e12 + 3e14 − 22e16 + 11e18 − 32e20 + 49e22 −e24 + 69e26 + 15e28 + 43e30 + 8e32 + 19e34 − 2e36 + e38 − e40 ) +b13 (e10 − e12 + e14 + e16 + 18e18 + 11e20 + 35e22 + 13e24 + 35e26 +11e28 + 18e30 + e32 + 4e34 − e36 + e38 ), R3 (b, e) = −e + e3 + e5 − 3e7 − 16e9 − 38e11 − 45e13 − 46e15 − 56e17 −50e19 − 32e21 − 14e23 − e27 − 2e29 + b(2e − e3 − e5 − 30e7 − 35e9 −87e11 − 44e13 − 95e15 − 44e17 − 33e19 + 27e21 + 15e23 + 22e25 − 5e27 +3e29 + 4e31 ) + b2 (e − 2e3 − 14e5 − 32e7 − 48e9 − 64e11 − 46e13 −73e15 + 41e17 + 49e19 + 101e21 + 45e23 + 36e25 − e27 + 12e29 − 3e31 −2e33 ) + b3 (−3e3 − 13e5 − 31e7 − 43e9 − 45e11 − 10e13 + 77e15 + 156e17 +197e19 + 174e21 + 105e23 + 42e25 + 7e27 − 5e29 − 5e31 + e33 ) + b4 (−2e3 −6e5 − 31e7 − 23e9 − 46e11 + 110e13 + 113e15 + 271e17 + 189e19 + 148e21 −16e23 − 19e25 − 61e27 − 10e29 − 6e31 − 7e33 ) + b5 (−e3 − 5e5 − 20e7

Appendix 1

167

−23e9 + 20e11 + 115e13 + 132e15 + 252e17 + 92e19 + 61e21 − 97e23 − 75e25 −110e27 − 17e29 − 33e31 + 5e33 + 6e35 ) + b6 (−4e5 − 23e7 + 14e9 + 33e11 +122e13 + 144e15 + 165e17 + 40e19 − 26e21 − 159e23 − 158e25 − 114e27 −48e29 − 5e31 + 17e33 + 3e35 − e37 ) + b7 (−4e5 + e7 + 14e9 + 19e11 +109e13 + 69e15 + 126e17 − 49e19 − 116e21 − 252e23 − 138e25 − 122e27 +4e29 + 13e31 + 16e33 + 8e35 ) + b8 (4e5 − e7 + e9 + 31e11 + 52e13 + 31e15 +17e17 − 147e19 − 199e21 − 230e23 − 154e25 − 84e27 + 24e29 + 6e31 +33e33 + 11e35 + e37 ) + b9 (−6e7 + 21e9 + 4e11 + 28e13 − 63e15 − 61e17 −224e19 − 145e21 − 207e23 − 58e25 − 18e27 + 35e29 + 35e31 + 38e33 +16e35 + 2e37 − e39 ) + b10 (2e7 − 8e11 + e13 − 51e15 − 36e17 − 122e19 −33e21 − 59e23 + 84e25 + 23e27 + 97e29 + 44e31 + 48e33 + 11e35 + e37 −2e39 ) + b11 (−2e7 + 3e11 − 8e13 − 24e15 − 26e17 − 21e19 + 20e21 +58e23 + 56e25 + 84e27 + 82e29 + 50e31 + 28e33 + 2e35 − e37 + e39 ) +b12 (4e9 − 6e11 − 2e13 − 15e15 + 26e17 + 5e19 + 73e21 + 27e23 + 80e25 +49e27 + 45e29 + 11e31 + 6e33 − 3e35 + 2e39 ) + b13 (−2e11 + 4e13 + e15 +15e17 − 8e19 + 21e21 − 18e23 + 26e25 − 27e27 + 6e29 − 19e31 + 7e33 −6e35 + 2e37 − 2e39 ), R4 (b, e) = 3e2 − 2e4 − 2e6 − 9e8 + 8e10 − 17e12 + 11e14 − 43e16 +26e18 − e20 + 25e22 − 6e24 + 6e26 − 3e28 + 4e30 + b(−2e2 − e4 −12e6 + 7e8 − 32e10 − 6e12 − 36e14 + 3e16 + 54e18 + 14e20 + 12e22 −7e24 + 10e26 − 2e28 + 6e30 − 8e32 ) + b2 (−2e2 − 4e4 − 3e6 − 22e8 −39e10 − 13e12 − 11e14 + 89e16 + 26e18 + 25e20 − 20e22 + 9e24 −22e26 − 14e30 − 3e32 + 4e34 ) + b3 (−4e4 − 16e6 − 30e8 − 10e10 −12e12 + 92e14 + 27e16 + 98e18 − 37e20 + 10e22 − 75e24 − 12e26 −33e28 + 2e32 ) + b4 (e2 − 8e4 − 22e6 − 2e8 − 20e10 + 92e12 + 44e14 +88e16 + 32e18 − 72e20 − 49e22 − 54e24 − 19e26 − 32e28 + 17e30 −12e32 + 16e34 ) + b5 (−5e4 − 10e6 − 16e8 + 27e10 + 61e12 + 59e14 +96e16 − 31e18 − 71e20 − 89e22 − 54e24 − 54e26 + 46e28 + 7e30 + 32e32 +10e34 − 8e36 ) + b6 (−2e4 − 13e6 + 14e8 + 10e10 + 49e12 + 73e14 + 35e16 −14e18 − 113e20 − 84e22 − 105e24 + 48e26 + 22e28 + 62e30 + 23e32 − e34 −4e36 ) + b7 (−2e4 + 2e6 − e8 + 8e10 + 81e12 + 25e14 + 32e16 − 99e18 −108e20 − 101e22 + 18e24 + 17e26 + 67e28 + 58e30 + 4e32 + 11e34 −10e36 − 2e38 ) + b8 (−5e6 + 39e10 + 28e12 + 20e14 − 39e16 − 77e18 −133e20 + 10e22 − 29e24 + 68e26 + 77e28 + 33e30 + 30e32 − 2e34 − 15e36 −5e38 ) + b9 (17e8 − 8e10 + 13e12 − 34e14 − 13e16 − 90e18 − 33e20 − 22e22 +2e24 + 88e26 + 38e28 + 78e30 + 4e32 − 10e34 − 28e36 − 2e38 ) + b10 (e6 −2e8 + 6e12 − 39e14 − 20e16 − 69e18 + 22e20 − 23e22 + 88e24 + 7e26 +92e28 − 2e30 − 10e32 − 34e34 − 14e36 − 3e38 ) + b11 (2e8 + 2e10 − 14e12 −13e14 − 14e16 + 5e18 − 14e20 + 55e22 − e24 + 73e26 − 20e30 − 29e32 −17e34 − 10e36 − 4e38 − e40 ) + b12 (−7e10 + 2e12 − 3e14 + 16e16 − 12e18 +30e20 + 13e22 + 20e24 + 17e26 − 30e28 − 8e30 − 22e32 − 3e34 − 12e36 +3e38 − 4e40 ) + b13 (4e12 − 2e14 + 7e16 − 5e18 + 27e20 − 13e22 + 20e24 −29e26 + 14e28 − 17e30 + 5e32 − 14e34 + 3e36 − e38 + e40 ), R5 (b, e) = −3e3 − 2e5 − 13e9 − 38e11 − 30e13 − 40e15 − 3e17 − 26e19 −5e21 − 6e23 + 16e25 − e27 + 2e29 − 2e31 + b(e − 3e3 − 8e7 − 35e9 −20e11 + 8e13 + 26e15 + 64e17 + 26e19 + 53e21 + 36e23 + 19e25 − 18e27

168

Appendix 1

+2e29 − 4e31 + 4e33 ) + b2 (−4e5 − 22e7 − 8e9 + 37e11 + 60e13 + 81e15 +78e17 + 56e19 + 57e21 + 15e23 − 29e25 − 13e27 − 5e29 − e31 + 2e33 −2e35 ) + b3 (−3e3 − 8e5 − 3e7 + 19e9 + 49e11 + 103e13 + 101e15 + 118e17 +36e19 + 31e21 − 60e23 − 29e25 − 47e27 − 5e29 − 3e31 + 3e33 ) + b4 (−5e3 +4e5 + 4e7 + 8e9 + 63e11 + 76e13 + 54e15 + 29e17 − 109e19 − 120e21 −169e23 − 115e25 − 64e27 + 35e29 + 13e33 − 6e35 ) + b5 (−e3 + e5 − 6e7 +26e9 + 59e11 + 38e13 + 3e15 − 82e17 − 204e19 − 143e21 − 159e23 −68e25 + 28e27 + 28e29 + 16e31 + 11e33 − 3e35 + 3e37 ) + b6 (−3e5 + 2e7 +26e9 + 42e11 + 42e13 − 33e15 − 130e17 − 192e19 − 134e21 − 117e23 +42e25 + 40e27 + 75e29 + 32e31 + 12e33 − 6e35 ) + b7 (17e9 + 30e11 −16e13 − 90e15 − 123e17 − 165e19 − 110e21 − 20e23 + 70e25 + 120e27 +119e29 + 22e31 + 11e33 − 13e35 − 3e37 ) + b8 (2e5 + 15e9 + 2e11 − 49e13 −78e15 − 109e17 − 145e19 + 3e21 − 4e23 + 131e25 + 143e27 + 92e29 +23e31 + e33 − 23e35 − 5e37 + e39 ) + b9 (−2e7 + 5e9 − 7e11 − 33e13 −67e15 − 94e17 − 43e19 + 62e21 + 92e23 + 222e25 + 140e27 + 88e29 − 14e31 −22e33 − 25e35 + 4e37 − 4e39 ) + b10 (2e9 − 6e11 − 17e13 − 7e15 + 32e17 +77e19 + 139e21 + 187e23 + 172e25 + 104e27 + 9e29 − 34e31 − 25e33 −14e35 − 10e37 − 5e39 ) + b11 (−e7 + e9 − 5e11 − 6e13 + 12e15 + 25e17 +52e19 + 110e21 + 61e23 + 55e25 − 13e27 − 30e29 − 37e31 − 28e33 − 34e35 −11e37 ) + b12 (2e9 − 2e11 + 2e13 + 10e15 + 19e17 + 40e19 + 30e21 − 16e23 −7e25 − 60e27 − 49e29 − 59e31 − 36e33 − 26e35 + 2e37 − 3e39 + 2e41 ) +b13 (−e11 + e13 + e15 − 3e17 − 16e19 − 38e21 − 45e23 − 46e25 − 56e27 −50e29 − 32e31 − 14e33 − e37 − 2e39 ), R6 (b, e) = 2e2 − 2e4 + e6 − 16e8 + 6e10 + 5e12 + 26e14 + 3e16 +40e18 + 30e20 + 38e22 + 13e24 + 2e28 + 3e30 + b(−3e2 − 7e6 + 12e10 +36e12 + 9e14 + 63e16 + 48e18 + 34e20 − e22 − 19e24 − 17e26 + 8e28 −6e30 − 6e32 ) + b2 (−2e2 − 3e4 − e6 + 6e8 + 19e10 + 19e12 + 35e14 +94e16 − 10e18 + 3e20 − 72e22 − 38e24 − 33e26 − 6e28 − 20e30 + 6e32 +3e34 ) + b3 (−3e4 − 4e8 + 29e10 + 31e12 + 82e14 − 15e16 − 22e18 − 120e20 −114e22 − 110e24 − 50e26 − 19e28 + 7e30 + 8e32 − 2e34 ) + b4 (−e4 − 11e6 +4e8 + 39e10 + 70e12 − 13e14 − 12e16 − 152e18 − 154e20 − 122e22 − 36e24 +10e26 + 50e28 + 5e30 + 9e32 + 12e34 ) + b5 (−3e4 − 7e6 + 12e8 + 50e10 −5e14 − 48e16 − 179e18 − 67e20 − 114e22 + 17e24 + 46e26 + 91e28 + 20e30 +52e32 − 8e34 − 8e36 ) + b6 (−3e4 + 22e8 + 8e10 + 7e12 − 19e14 − 93e16 −93e18 − 96e20 − 65e22 + 58e24 + 120e26 + 98e28 + 69e30 + 8e32 − 20e34 −e36 ) + b7 (2e6 − e8 + 14e10 + 14e12 − 48e14 − 36e16 − 144e18 − 47e20 +29e22 + 161e24 + 112e26 + 120e28 − 3e32 − 10e34 − 11e36 − e38 ) +b8 (−6e6 + 7e8 + 24e10 − 32e12 − 2e14 − 73e16 − 65e18 + 33e20 + 85e22 +136e24 + 159e26 + 78e28 − 3e30 + 13e32 − 40e34 − 11e36 − e38 ) + b9 (e6 +11e8 − 16e10 − 14e12 − 26e14 − 10e16 + 9e18 + 98e20 + 61e22 + 170e24 +78e26 + 23e28 + 4e30 − 32e32 − 36e34 − 16e36 − 5e38 + 2e40 ) + b10 (−2e8 +e10 − e12 − 13e14 + 17e16 − e18 + 90e20 + 35e22 + 86e24 − 51e26 + 14e28 −82e30 − 33e32 − 50e34 − 12e36 − e38 + 3e40 ) + b11 (3e8 − e10 − 13e12 +10e14 + e16 + 29e18 + 20e20 + 10e22 − 16e24 + 11e26 − 66e28 − 68e30

Appendix 1 −49e32 − 29e34 + 6e36 + 3e38 − 2e40 ) + b12 (−6e10 + 4e12 + 9e14 + 8e16 −2e18 + 5e20 − 30e22 + 10e24 − 59e26 − 56e28 − 31e30 − 11e32 + 4e34 +7e36 − e38 − 2e40 ) + b13 (3e12 − 2e14 − 2e16 − 9e18 + 8e20 − 17e22 +11e24 − 43e26 + 26e28 − e30 + 25e32 − 6e34 + 6e36 − 3e38 + 4e40 ), R13−k (b, e) = −b13 e45 Rk (b−1 , e−1 ), (k = 0, 6).

169

Appendix 2

The Polynomials R2k (b, f ), that Define N1,5 (u, b, f ) R0 (b, f ) = 1 + f − f 3 + f 4 + 4f 5 + 11f 6 + 16f 7 + 17f 8 + 13f 9 +13f 10 + 13f 11 + 17f 12 + 16f 13 + 11f 14 + 4f 15 + f 16 − f 17 + f 19 +f 20 + b(−2f − f 2 + 5f 3 + 12f 4 + 15f 5 + 15f 6 + 11f 7 + 10f 8 +17f 9 + 27f 10 + 25f 11 + 16f 12 − f 13 − 9f 14 − 5f 15 + 2f 16 + 3f 17 +3f 18 − 2f 20 − 2f 21 ) + b2 (−f + 3f 2 + 6f 3 + 5f 4 − 3f 6 − 10f 7 −13f 8 − 15f 9 − 18f 10 − 32f 11 − 41f 12 − 51f 13 − 43f 14 − 29f 15 −18f 16 − 11f 17 − 3f 18 − 2f 19 − 2f 20 − f 21 + f 22 ) + b3 (3f 2 + f 3 −5f 4 − 11f 5 − 11f 6 − 15f 7 − 20f 8 − 35f 9 − 58f 10 − 85f 11 − 81f 12 −61f 13 − 21f 14 − f 15 − 7f 17 − 7f 18 − 8f 19 − 2f 20 + 3f 21 + 4f 22 ) +b4 (2f 2 − f 3 − 4f 4 − 6f 5 − 9f 6 − 18f 7 − 21f 8 − 21f 9 − 23f 10 −22f 11 − 6f 12 + 6f 13 + 22f 14 + 23f 15 + 21f 16 + 21f 17 + 18f 18 +9f 19 + 6f 20 + 4f 21 + f 22 − 2f 23 ) + b5 (−4f 3 − 3f 4 + 2f 5 + 8f 6 +7f 7 + 7f 8 + f 10 + 21f 11 + 61f 12 + 81f 13 + 85f 14 + 58f 15 + 35f 16 +20f 17 + 15f 18 + 11f 19 + 11f 20 + 5f 21 − f 22 − 3f 23 ) + b6 (−f 3 + f 4 +2f 5 + 2f 6 + 3f 7 + 11f 8 + 18f 9 + 29f 10 + 43f 11 + 51f 12 + 41f 13 +32f 14 + 18f 15 + 15f 16 + 13f 17 + 10f 18 + 3f 19 − 5f 21 − 6f 22 − 3f 23 +f 24 ) + b7 (2f 4 + 2f 5 − 3f 7 − 3f 8 − 2f 9 + 5f 10 + 9f 11 + f 12 − 16f 13 −25f 14 − 27f 15 − 17f 16 − 10f 17 − 11f 18 − 15f 19 − 15f 20 − 12f 21 −5f 22 + f 23 + 2f 24 ) + b8 (−f 5 − f 6 + f 8 − f 9 − 4f 10 − 11f 11 − 16f 12 −17f 13 − 13f 14 − 13f 15 − 13f 16 − 17f 17 − 16f 18 − 11f 19 − 4f 20 −f 21 + f 22 − f 24 − f 25 ), R2 (b, f ) = −2f − f 2 + 5f 3 + 12f 4 + 15f 5 + 15f 6 + 11f 7 + 10f 8 +17f 9 + 27f 10 + 25f 11 + 16f 12 − f 13 − 9f 14 − 5f 15 + 2f 16 + 3f 17 +3f 18 − 2f 20 − 2f 21 + b(f + 9f 2 + 12f 3 + 3f 4 − 4f 5 − 3f 6 + 11f 7 +28f 8 + 34f 9 + 10f 10 − 22f 11 − 36f 12 − 29f 13 − 3f 14 + 12f 15 − f 16 −12f 17 − 11f 18 − 7f 19 + 4f 21 + 4f 22 ) + b2 (2f + 6f 2 − 4f 3 − 15f 4 −13f 5 − 7f 6 − 6f 7 − 20f 8 − 53f 9 − 92f 10 − 107f 11 − 84f 12 − 44f 13 +f 14 + 9f 15 − 3f 16 + f 17 + 4f 18 + f 19 + 3f 20 + 4f 21 + 2f 22 − 2f 23 ) +b3 (f − f 2 − 11f 3 − 10f 4 + f 5 − 3f 6 − 26f 7 − 64f 8 − 89f 9 − 85f 10 −35f 11 + 31f 12 + 59f 13 + 46f 14 − 3f 15 − 18f 16 + 9f 17 + 25f 18 +24f 19 + 19f 20 + 5f 21 − 6f 22 − 8f 23 ) + b4 (−3f 2 − 9f 3 + 9f 5 +f 6 − 8f 7 − 13f 8 − 4f 9 + 15f 10 + 65f 11 + 99f 12 + 99f 13 + 75f 14 +42f 15 + 35f 16 + 34f 17 + 6f 18 − 6f 19 − 6f 20 − 8f 21 − 8f 22 −2f 23 + 4f 24 ) + b5 (−2f 2 + 12f 4 + 11f 5 − 3f 6 − 6f 7 + 9f 8 + 45f 9 +97f 10 + 139f 11 + 118f 12 + 52f 13 − f 14 − 20f 15 + 7f 16 + 16f 17 171

172

Appendix 2

−11f 19 − 19f 20 − 23f 21 − 12f 22 + 2f 23 + 6f 24 ) + b6 (4f 3 + 7f 4 − f 5 −7f 6 + 4f 7 + 19f 8 + 37f 9 + 45f 10 + 31f 11 − 20f 12 − 51f 13 − 49f 14 −27f 15 − 9f 16 − 20f 17 − 40f 18 − 39f 19 − 29f 20 − 14f 21 + 4f 22 +12f 23 + 6f 24 − 2f 25 ) + b7 (f 3 − f 4 − 7f 5 − 6f 6 + 2f 7 + 4f 8 − 2f 9 −23f 10 − 59f 11 − 88f 12 − 74f 13 − 45f 14 − 25f 15 − 28f 16 − 46f 17 −47f 18 − 19f 19 + 6f 20 + 21f 21 + 21f 22 + 8f 23 − 5f 24 − 5f 25 ) +b8 (−2f 4 − 2f 5 + 2f 6 + 4f 7 − 2f 8 − 10f 9 − 20f 10 − 24f 11 − 12f 12 +6f 13 + 8f 14 − 8f 16 − 6f 17 + 12f 18 + 24f 19 + 20f 20 + 10f 21 + 2f 22 −4f 23 − 2f 24 + 2f 25 + 2f 26 ) + b9 (f 5 + f 6 − f 8 + f 9 + 4f 10 + 11f 11 +16f 12 + 17f 13 + 13f 14 + 13f 15 + 13f 16 + 17f 17 + 16f 18 + 11f 19 +4f 20 + f 21 − f 22 + f 24 + f 25 ), R4 (b, f ) = −f + 3f 2 + 6f 3 + 5f 4 − 3f 6 − 10f 7 − 13f 8 − 15f 9 − 18f 10 −32f 11 − 41f 12 − 51f 13 − 43f 14 − 29f 15 − 18f 16 − 11f 17 − 3f 18 −2f 19 − 2f 20 − f 21 + f 22 + b(2f + 6f 2 − 4f 3 − 15f 4 − 13f 5 − 7f 6 −6f 7 − 20f 8 − 53f 9 − 92f 10 − 107f 11 − 84f 12 − 44f 13 + f 14 + 9f 15 −3f 16 + f 17 + 4f 18 + f 19 + 3f 20 + 4f 21 + 2f 22 − 2f 23 ) + b2 (f − 3f 2 −17f 3 − 11f 4 − 2f 5 − 13f 6 − 44f 7 − 69f 8 − 72f 9 − 37f 10 + 18f 11 +81f 12 + 114f 13 + 126f 14 + 96f 15 + 85f 16 + 83f 17 + 50f 18 + 19f 19 +9f 20 + 5f 21 − 3f 23 + f 24 ) + b3 (−5f 2 − 8f 3 + 7f 4 + 5f 5 − 19f 6 −41f 7 − 28f 8 + 24f 9 + 104f 10 + 190f 11 + 251f 12 + 233f 13 + 175f 14 +95f 15 + 74f 16 + 52f 17 + 13f 18 + f 19 + 4f 20 − 4f 21 − 10f 22 − 5f 23 +4f 24 ) + b4 (−2f 2 + 2f 3 + 13f 4 − 4f 5 − 18f 6 + 9f 7 + 72f 8 + 131f 9 +165f 10 + 175f 11 + 133f 12 + 64f 13 + 14f 14 − 17f 15 − 23f 16 − 65f 17 −98f 18 − 67f 19 − 37f 20 − 23f 21 − 12f 22 + 2f 23 + 5f 24 − 2f 25 ) +b5 (8f 3 + 11f 4 − 8f 5 − 6f 6 + 23f 7 + 62f 8 + 84f 9 + 94f 10 +52f 11 − 48f 12 − 152f 13 − 196f 14 − 197f 15 − 161f 16 − 154f 17 −115f 18 − 66f 19 − 45f 20 − 29f 21 − 4f 22 + 11f 23 + 5f 24 − 3f 25 ) +b6 (f 3 − 7f 4 − 11f 5 + 15f 6 + 41f 7 + 33f 8 − 13f 9 − 74f 10 − 140f 11 −176f 12 − 151f 13 − 114f 14 − 107f 15 − 121f 16 − 130f 17 − 82f 18 −21f 19 + 13f 20 + 30f 21 + 32f 22 + 17f 23 − 3f 24 − 6f 25 + f 26 ) +b7 (−4f 4 − 3f 5 + 5f 6 − 2f 7 − 23f 8 − 46f 9 − 64f 10 − 80f 11 − 66f 12 −30f 13 − 10f 14 + 10f 16 + 30f 17 + 66f 18 + 80f 19 + 64f 20 + 46f 21 +23f 22 + 2f 23 − 5f 24 + 3f 25 + 4f 26 ) + b8 (5f 5 + 5f 6 − 8f 7 − 21f 8 −21f 9 − 6f 10 + 19f 11 + 47f 12 + 46f 13 + 28f 14 + 25f 15 + 45f 16 +74f 17 + 88f 18 + 59f 19 + 23f 20 + 2f 21 − 4f 22 − 2f 23 + 6f 24 + 7f 25 +f 26 − f 27 ) + b9 (−2f 6 − f 7 + 5f 8 + 12f 9 + 15f 10 + 15f 11 + 11f 12 +10f 13 + 17f 14 + 27f 15 + 25f 16 + 16f 17 − f 18 − 9f 19 − 5f 20 + 2f 21 +3f 22 + 3f 23 − 2f 25 − 2f 26 ), R6 (b, f ) = 3f 2 + f 3 − 5f 4 − 11f 5 − 11f 6 − 15f 7 − 20f 8 − 35f 9 −58f 10 − 85f 11 − 81f 12 − 61f 13 − 21f 14 − f 15 − 7f 17 − 7f 18 − 8f 19 −2f 20 + 3f 21 + 4f 22 + b(f − f 2 − 11f 3 − 10f 4 + f 5 − 3f 6 − 26f 7 −64f 8 − 89f 9 − 85f 10 − 35f 11 + 31f 12 + 59f 13 + 46f 14 − 3f 15 −18f 16 + 9f 17 + 25f 18 + 24f 19 + 19f 20 + 5f 21 − 6f 22 − 8f 23 ) +b2 (−5f 2 − 8f 3 + 7f 4 + 5f 5 − 19f 6 − 41f 7 − 28f 8 + 24f 9 +104f 10 + 190f 11 + 251f 12 + 233f 13 + 175f 14 + 95f 15 + 74f 16

Appendix 2

173

+52f 17 + 13f 18 + f 19 + 4f 20 − 4f 21 − 10f 22 − 5f 23 + 4f 24 ) +b3 (−3f 2 + 3f 3 + 8f 4 − 15f 5 − 33f 6 − 6f 7 + 73f 8 + 167f 9 + 226f 10 +235f 11 + 162f 12 + 29f 13 − 36f 14 − 27f 15 + 45f 16 + 29f 17 − 20f 18 −43f 19 − 51f 20 − 51f 21 − 22f 22 + 9f 23 + 16f 24 ) + b4 (7f 3 + 4f 4 −16f 5 − 3f 6 + 45f 7 + 86f 8 + 92f 9 + 67f 10 + 26f 11 − 82f 12 − 174f 13 −199f 14 − 183f 15 − 160f 16 − 172f 17 − 128f 18 − 50f 19 − 22f 20 − 7f 21 +14f 22 + 23f 23 + 6f 24 − 8f 25 ) + b5 (3f 3 − 8f 4 − 12f 5 + 23f 6 + 56f 7 +53f 8 − 80f 10 − 194f 11 − 290f 12 − 276f 13 − 176f 14 − 110f 15 − 100f 16 −137f 17 − 91f 18 − 41f 19 + 2f 20 + 46f 21 + 65f 22 + 35f 23 − 6f 24 −13f 25 ) + b6 (−6f 4 + 7f 5 + 25f 6 + 9f 7 − 37f 8 − 79f 9 − 104f 10 −112f 11 − 86f 12 − 13f 13 + 16f 14 − 16f 16 + 13f 17 + 86f 18 +112f 19 + 104f 20 + 79f 21 + 37f 22 − 9f 23 − 25f 24 − 7f 25 + 6f 26 ) +b7 (−f 4 + 6f 5 + 3f 6 − 17f 7 − 32f 8 − 30f 9 − 13f 10 + 21f 11 +82f 12 + 130f 13 + 121f 14 + 107f 15 + 114f 16 + 151f 17 + 176f 18 +140f 19 + 74f 20 + 13f 21 − 33f 22 − 41f 23 − 15f 24 + 11f 25 + 7f 26 −f 27 ) + b8 (2f 5 − 6f 6 − 12f 7 − 4f 8 + 14f 9 + 29f 10 + 39f 11 + 40f 12 +20f 13 + 9f 14 + 27f 15 + 49f 16 + 51f 17 + 20f 18 − 31f 19 − 45f 20 −37f 21 − 19f 22 − 4f 23 + 7f 24 + f 25 − 7f 26 − 4f 27 ) + b9 (−f 6 +3f 7 + 6f 8 + 5f 9 − +3f 11 − 10f 12 − 13f 13 − 15f 14 − 18f 15 −32f 16 − 41f 17 − 51f 18 − 43f 19 − 29f 20 − 18f 21 − 11f 22 − 3f 23 −2f 24 − 2f 25 − f 26 + f 27 ), R8 (b, f ) = 2f 2 − f 3 − 4f 4 − 6f 5 − 9f 6 − 18f 7 − 21f 8 − 21f 9 − 23f 10 −22f 11 − 6f 12 + 6f 13 + 22f 14 + 23f 15 + 21f 16 + 21f 17 + 18f 18 +9f 19 + 6f 20 + 4f 21 + f 22 − 2f 23 + b(−3f 2 − 9f 3 + 9f 5 + f 6 − 8f 7 −13f 8 − 4f 9 + 15f 10 + 65f 11 + 99f 12 + 99f 13 + 75f 14 + 42f 15 + 35f 16 +34f 17 + 6f 18 − 6f 19 − 6f 20 − 8f 21 − 8f 22 − 2f 23 + 4f 24 ) + b2 (−2f 2 +2f 3 + 13f 4 − 4f 5 − 18f 6 + 9f 7 + 72f 8 + 131f 9 + 165f 10 + 175f 11 +133f 12 + 64f 13 + 14f 14 − 17f 15 − 23f 16 − 65f 17 − 98f 18 − 67f 19 −37f 20 − 23f 21 − 12f 22 + 2f 23 + 5f 24 − 2f 25 ) + b3 (7f 3 + 4f 4 − 16f 5 −3f 6 + 45f 7 + 86f 8 + 92f 9 + 67f 10 + 26f 11 − 82f 12 − 174f 13 − 199f 14 −183f 15 − 160f 16 − 172f 17 − 128f 18 − 50f 19 − 22f 20 − 7f 21 + 14f 22 +23f 23 + 6f 24 − 8f 25 ) + b4 (f 3 − 8f 4 − 9f 5 + 33f 6 + 61f 7 + 20f 8 −67f 9 − 145f 10 − 201f 11 − 272f 12 − 243f 13 − 200f 14 − 178f 15 −181f 16 − 141f 17 − 16f 18 + 81f 19 + 77f 20 + 72f 21 + 55f 22 + 25f 23 −10f 24 − 9f 25 + 4f 26 ) + b5 (−8f 4 + +f 5 + 24f 6 + 7f 7 − 42f 8 − 89f 9 −123f 10 − 182f 11 − 209f 12 − 130f 13 − 55f 14 + 55f 16 + 130f 17 +209f 18 + 182f 19 + 123f 20 + 89f 21 + 42f 22 − 7f 23 − 24f 24 − f 25 +8f 26 ) + b6 (13f 5 + 6f 6 − 35f 7 − 65f 8 − 46f 9 − 2f 10 + 41f 11 +91f 12 + 137f 13 + 100f 14 + 110f 15 + 176f 16 + 276f 17 + 290f 18 +194f 19 + 80f 20 − 53f 22 − 56f 23 − 23f 24 + 12f 25 + 8f 26 − 3f 27 ) +b7 (3f 5 − 5f 6 − 11f 7 + 4f 8 + 29f 9 + 45f 10 + 66f 11 + 115f 12 +154f 13 + 161f 14 + 197f 15 + 196f 16 + 152f 17 + 48f 18 − 52f 19 −94f 20 − 84f 21 − 62f 22 − 23f 23 + 6f 24 + 8f 25 − 11f 26 − 8f 27 ) +b8 (−6f 6 − 2f 7 + 12f 8 + 23f 9 + 19f 10 + 11f 11 − 16f 13 − 7f 14

174

Appendix 2 +20f 15 + f 16 − 52f 17 − 118f 18 − 139f 19 − 97f 20 − 45f 21 − 9f 22 +6f 23 + 3f 24 − 11f 25 − 12f 26 + 2f 28 ) + b9 (3f 7 + f 8 − 5f 9 − 11f 10 −11f 11 − 15f 12 − 20f 13 − 35f 14 − 58f 15 − 85f 16 − 81f 17 − 61f 18 −21f 19 − f 20 − 7f 22 − 7f 23 − 8f 24 − 2f 25 + 3f 26 + 4f 27 ), R16−2k (b, f ) = −b9 f 30 R2k (b−1 , f −1 ), (k = 0, 3).

Appendix 3

Polynomials that Define the Quantity G1 for the System s(1, 2). Expressions of Focus Pseudo-Quantities G1,i and σ1,i , B1,i (i = 0, 1, 2, 3, 4) G1,0 = −17c3 d2 e3 g 2 + 11cd3 e4 g 2 + 34c4 de2 f g 2 − 118c2 d2 e3 f g 2 +15d3 e4 f g 2 − 17c5 ef 2 g 2 + 215c3 de2 f 2 g 2 − 200cd2 e3 f 2 g 2 −108c4 ef 3 g 2 + 403c2 de2 f 3 g 2 − 91d2 e3 f 3 g 2 − 224c3 ef 4 g 2 +247cde2 f 4 g 2 − 174c2 ef 5 g 2 + 29de2 f 5 g 2 − 47cef 6 g 2 − 6ef 7 g 2 +34c2 d2 e4 gh − 4d3 e5 gh − 92c3 de3 f gh + 152cd2 e4 f gh + 58c4 e2 f 2 gh −400c2 de3 f 2 gh + 118d2 e4 f 2 gh + 276c3 e2 f 3 gh − 400cde3 f 3 gh +354c2 e2 f 4 gh − 80de3 f 4 gh + 152ce2 f 5 gh + 24e2 f 6 gh − 24cd2 e5 h2 +72c2 de4 f h2 − 32d2 e5 f h2 − 72c3 e3 f 2 h2 + 132cde4 f 2 h2 −172c2 e3 f 3 h2 + 44de4 f 3 h2 − 116ce3 f 4 h2 − 24e3 f 5 h2 + 12c3 de4 gk −12cd2 e5 gk − 12c4 e3 f gk + 72c2 de4 f gk − 16d2 e5 f gk − 72c3 e3 f 2 gk +102cde4 f 2 gk − 122c2 e3 f 3 gk + 34de4 f 3 gk − 70ce3 f 4 gk − 12e3 f 5 gk −12c2 de5 hk + 4d2 e6 hk + 36c3 e4 f hk − 52cde5 f hk + 120c2 e4 f 2 hk −32de5 f 2 hk + 104ce4 f 3 hk + 24e4 f 4 hk − 6c3 e5 k 2 + 5cde6 k 2 −23c2 e5 f k 2 + 5de6 f k 2 − 23ce5 f 2 k 2 − 6e5 f 3 k 2 + 9c4 d2 e2 gl −13c2 d3 e3 gl − 4d4 e4 gl − 18c5 def gl + 65c3 d2 e2 f gl − 16cd3 e3 f gl +9c6 f 2 gl − 103c4 def 2 gl + 90c2 d2 e2 f 2 gl − 11d3 e3 f 2 gl + 51c5 f 3 gl −140c3 def 3 gl + 7cd2 e2 f 3 gl + 76c4 f 4 gl + 26c2 def 4 gl − 31d2 e2 f 4 gl −10c3 f 5 gl + 110cdef 5 gl − 79c2 f 6 gl + 29def 6 gl − 41cf 7 gl − 6f 8 gl −12c3 d2 e3 hl + 26cd3 e4 hl + 48c4 de2 f hl − 122c2 d2 e3 f hl + 22d3 e4 f hl −36c5 ef 2 hl + 250c3 de2 f 2 hl − 104cd2 e3 f 2 hl − 166c4 ef 3 hl +248c2 de2 f 3 hl − 2d2 e3 f 3 hl − 190c3 ef 4 hl + 40cde2 f 4 hl − 38c2 ef 5 hl −10de2 f 5 hl + 34cef 6 hl + 12ef 7 hl − 12c4 de3 kl + 11c2 d2 e4 kl +12c5 e2 f kl − 70c3 de3 f kl + 16cd2 e4 f kl + 65c4 e2 f 2 kl − 109c2 de3 f 2 kl +5d2 e4 f 2 kl + 105c3 e2 f 3 kl − 58cde3 f 3 kl + 53c2 e2 f 4 kl − 7de3 f 4 kl −5ce2 f 5 kl − 6e2 f 6 kl + 3c3 d3 e2 l2 − 5cd4 e3 l2 − 6c4 d2 ef l2 + 20c2 d3 e2 f l2 −5d4 e3 f l2 + 3c5 df 2 l2 − 37c3 d2 ef 2 l2 + 26cd3 e2 f 2 l2 + 22c4 df 3 l2 −73c2 d2 ef 3 l2 + 9d3 e2 f 3 l2 + 58c3 df 4 l2 − 59cd2 ef 4 l2 + 68c2 df 5 l2 −17d2 ef 5 l2 + 35cdf 6 l2 + 6df 7 l2 − 12c3 d2 e3 gm + 2cd3 e4 gm +24c4 de2 f gm − 90c2 d2 e3 f gm − 2d3 e4 f gm − 12c5 ef 2 gm +138c3 de2 f 2 gm − 152cd2 e3 f 2 gm − 62c4 ef 3 gm + 200c2 de2 f 3 gm −82d2 e3 f 3 gm − 70c3 ef 4 gm + 32cde2 f 4 gm + 50c2 ef 5 gm 175

176

Appendix 3

−58de2 f 5 gm + 82cef 6 gm + 12ef 7 gm + 20c2 d2 e4 hm − 40c3 de3 f hm +96cd2 e4 f hm + 44c4 e2 f 2 hm − 212c2 de3 f 2 hm + 76d2 e4 f 2 hm +164c3 e2 f 3 hm − 152cde3 f 3 hm + 76c2 e2 f 4 hm + 20de3 f 4 hm −68ce2 f 5 hm − 24e2 f 6 hm + 12c3 de4 km − 6cd2 e5 km − 24c4 e3 f km +70c2 de4 f km − 10d2 e5 f km − 88c3 e3 f 2 km + 80cde4 f 2 km −70c2 e3 f 3 km + 14de4 f 3 km + 10ce3 f 4 km + 12e3 f 5 km + 6c4 d2 e2 lm −4c2 d3 e3 lm − 4d4 e4 lm − 12c5 def lm + 64c3 d2 e2 f lm − 28cd3 e3 f lm +6c6 f 2 lm − 92c4 def 2 lm + 180c2 d2 e2 f 2 lm − 32d3 e3 f 2 lm + 32c5 f 3 lm −212c3 def 3 lm + 148cd2 e2 f 3 lm + 40c4 f 4 lm − 168c2 def 4 lm + 22d2 e2 f 4 lm −20c3 f 5 lm − 48cdef 5 lm − 46c2 f 6 lm − 12def 6 lm − 12cf 7 lm − 4c3 d2 e3 m2 +8c4 de2 f m2 − 20c2 d2 e3 f m2 − 8d3 e4 f m2 − 4c5 ef 2 m2 + 64c3 de2 f 2 m2 −48cd2 e3 f 2 m2 − 20c4 ef 3 m2 + 148c2 de2 f 3 m2 − 48d2 e3 f 3 m2 − 20c3 ef 4 m2 +100cde2 f 4 m2 + 20c2 ef 5 m2 + 24cef 6 m2 + 31c2 d2 e4 gn − 38c3 de3 f gn +96cd2 e4 f gn + 13c4 e2 f 2 gn − 125c2 de3 f 2 gn + 65d2 e4 f 2 gn + 41c3 e2 f 3 gn −58cde3 f 3 gn − 7c2 e2 f 4 gn + 29de3 f 4 gn − 41ce2 f 5 gn − 6e2 f 6 gn −12c3 de4 hn − 46cd2 e5 hn + 78c2 de4 f hn − 50d2 e5 f hn − 56c3 e3 f 2 hn +88cde4 f 2 hn − 38c2 e3 f 3 hn − 10de4 f 3 hn + 34ce3 f 4 hn + 12e3 f 5 hn +6c4 e4 kn − 17c2 de5 kn + 4d2 e6 kn + 29c3 e4 f kn − 32cde5 f kn + 28c2 e4 f 2 kn −7de5 f 2 kn − 5ce4 f 3 kn − 6e4 f 4 kn − 22c3 d2 e3 ln + 32cd3 e4 ln + 32c4 de2 f ln −134c2 d2 e3 f ln + 28d3 e4 f ln − 10c5 ef 2 ln + 148c3 de2 f 2 ln − 116cd2 e3 f 2 ln −34c4 ef 3 ln + 126c2 de2 f 3 ln − 12d2 e3 f 3 ln − 2c3 ef 4 ln + 14cde2 f 4 ln +34c2 ef 5 ln + 12cef 6 ln + 14c2 d2 e4 mn + 4d3 e5 mn − 52c3 de3 f mn +88cd2 e4 f mn + 14c4 e2 f 2 mn − 204c2 de3 f 2 mn + 74d2 e4 f 2 mn + 40c3 e2 f 3 mn −128cde3 f 3 mn − 18c2 e2 f 4 mn + 12de3 f 4 mn − 36ce2 f 5 mn + 6c3 de4 n2 −31cd2 e5 n2 + 63c2 de4 f n2 − 27d2 e5 f n2 − 14c3 e3 f 2 n2 + 43cde4 f 2 n2 +2c2 e3 f 3 n2 − 6de4 f 3 n2 + 12ce3 f 4 n2 ; G1,1 = 2c4 d2 e2 g 2 + 9c2 d3 e3 g 2 − 4c5 def g 2 − 14c3 d2 e2 f g 2 + 41cd3 e3 f g 2 +2c6 f 2 g 2 − 5c4 def 2 g 2 − 102c2 d2 e2 f 2 g 2 + 36d3 e3 f 2 g 2 + 10c5 f 3 g 2 +57c3 def 3 g 2 − 129cd2 e2 f 3 g 2 + 10c4 f 4 g 2 + 121c2 def 4 g 2 − 35d2 e2 f 4 g 2 −10c3 f 5 g 2 + 65cdef 5 g 2 − 12c2 f 6 g 2 + 6def 6 g 2 − 22c3 d2 e3 gh − 12cd3 e4 gh +56c4 de2 f gh − 52c2 d2 e3 f gh − 16d3 e4 f gh − 34c5 ef 2 gh + 200c3 de2 f 2 gh −2cd2 e3 f 2 gh − 160c4 ef 3 gh + 144c2 de2 f 3 gh + 28d2 e3 f 3 gh − 198c3 ef 4 gh −24cde2 f 4 gh − 76c2 ef 5 gh − 12de2 f 5 gh − 12cef 6 gh + 24c2 d2 e4 h2 −72c3 de3 f h2 + 32cd2 e4 f h2 + 72c4 e2 f 2 h2 − 132c2 de3 f 2 h2 + 172c3 e2 f 3 h2 −44cde3 f 3 h2 + 116c2 e2 f 4 h2 + 24ce2 f 5 h2 − 6c4 de3 gk + 4c2 d2 e4 gk +6c5 e2 f gk − 32c3 de3 f gk + 40c4 e2 f 2 gk − 34c2 de3 f 2 gk − 8d2 e4 f 2 gk +70c3 e2 f 3 gk + 6cde3 f 3 gk + 38c2 e2 f 4 gk + 6de3 f 4 gk + 6ce2 f 5 gk +12c3 de4 hk − 4cd2 e5 hk − 36c4 e3 f hk + 52c2 de4 f hk − 120c3 e3 f 2 hk +32cde4 f 2 hk − 104c2 e3 f 3 hk − 24ce3 f 4 hk + 6c4 e4 k 2 − 5c2 de5 k 2 +23c3 e4 f k 2 − 5cde5 f k 2 + 23c2 e4 f 2 k 2 + 6ce4 f 3 k 2 + c3 d3 e2 gl + 4cd4 e3 gl +4c4 d2 ef gl − 4c2 d3 e2 f gl − 5c5 df 2 gl + 8c3 d2 ef 2 gl + 23cd3 e2 f 2 gl −14c4 df 3 gl − 53c2 d2 ef 3 gl + 20d3 e2 f 3 gl + 18c3 df 4 gl − 76cd2 ef 4 gl +56c2 df 5 gl − 23d2 ef 5 gl + 35cdf 6 gl + 6df 7 gl − 10c2 d3 e3 hl − 12c5 def hl +22c3 d2 e2 f hl − 6cd3 e3 f hl + 12c6 f 2 hl − 50c4 def 2 hl − 12c2 d2 e2 f 2 hl +50c5 f 3 hl + 8c3 def 3 hl − 26cd2 e2 f 3 hl + 34c4 f 4 hl + 64c2 def 4 hl − 38c3 f 5 hl

Appendix 3

177

+22cdef 5 hl − 46c2 f 6 hl − 12cf 7 hl + 6c5 de2 kl − 3c3 d2 e3 kl − 6c6 ef kl +30c4 de2 f kl − 33c5 ef 2 kl + 41c3 de2 f 2 kl + 3cd2 e3 f 2 kl − 53c4 ef 3 kl +18c2 de2 f 3 kl − 21c3 ef 4 kl + cde2 f 4 kl + 11c2 ef 5 kl + 6cef 6 kl + c2 d4 e2 l2 +4c3 d3 ef l2 − 3cd4 e2 f l2 − 5c4 d2 f 2 l2 + 24c2 d3 ef 2 l2 − 4d4 e2 f 2 l2 − 27c3 d2 f 3 l2 +31cd3 ef 3 l2 − 45c2 d2 f 4 l2 + 11d3 ef 4 l2 − 29cd2 f 5 l2 − 6d2 f 6 l2 + 6c4 d2 e2 gm +6c2 d3 e3 gm − 12c5 def gm + 30c3 d2 e2 f gm + 34cd3 e3 f gm + 6c6 f 2 gm −54c4 def 2 gm + 24d3 e3 f 2 gm + 30c5 f 3 gm − 40c3 def 3 gm − 18cd2 e2 f 3 gm +30c4 f 4 gm + 24c2 def 4 gm − 2d2 e2 f 4 gm − 30c3 f 5 gm + 14cdef 5 gm −36c2 f 6 gm − 12def 6 gm − 20c3 d2 e3 hm + 40c4 de2 f hm − 96c2 d2 e3 f hm −44c5 ef 2 hm + 212c3 de2 f 2 hm − 76cd2 e3 f 2 hm − 164c4 ef 3 hm +152c2 de2 f 3 hm − 76c3 ef 4 hm − 20cde2 f 4 hm + 68c2 ef 5 hm + 24cef 6 hm −12c4 de3 km + 6c2 d2 e4 km + 24c5 e2 f km − 70c3 de3 f km + 10cd2 e4 f km +88c4 e2 f 2 km − 80c2 de3 f 2 km + 70c3 e2 f 3 km − 14cde3 f 3 km − 10c2 e2 f 4 km −12ce2 f 5 km − 4c3 d3 e2 lm + 4cd4 e3 lm − 4c4 d2 ef lm − 4c2 d3 e2 f lm +8c5 df 2 lm − 28c3 d2 ef 2 lm + 8cd3 e2 f 2 lm + 52c4 df 3 lm − 48c2 d2 ef 3 lm +112c3 df 4 lm − 20cd2 ef 4 lm + 92c2 df 5 lm + 24cdf 6 lm + 4c4 d2 e2 m2 −8c5 def m2 + 20c3 d2 e2 f m2 + 8cd3 e3 f m2 + 4c6 f 2 m2 − 64c4 def 2 m2 +48c2 d2 e2 f 2 m2 + 20c5 f 3 m2 − 148c3 def 3 m2 + 48cd2 e2 f 3 m2 + 20c4 f 4 m2 −100c2 def 4 m2 − 20c3 f 5 m2 − 24c2 f 6 m2 − 19c3 d2 e3 gn − 16cd3 e4 gn +20c4 de2 f gn − 20c2 d2 e3 f gn − 16d3 e4 f gn − 7c5 ef 2 gn + 45c3 de2 f 2 gn +3cd2 e3 f 2 gn − 21c4 ef 3 gn + 6c2 de2 f 3 gn + 4d2 e3 f 3 gn + 7c3 ef 4 gn −13cde2 f 4 gn + 21c2 ef 5 gn + 6de2 f 5 gn + 12c4 de3 hn + 46c2 d2 e4 hn −78c3 de3 f hn + 50cd2 e4 f hn + 56c4 e2 f 2 hn − 88c2 de3 f 2 hn + 38c3 e2 f 3 hn +10cde3 f 3 hn − 34c2 e2 f 4 hn − 12ce2 f 5 hn − 6c5 e3 kn + 17c3 de4 kn −4cd2 e5 kn − 29c4 e3 f kn + 32c2 de4 f kn − 28c3 e3 f 2 kn + 7cde4 f 2 kn +5c2 e3 f 3 kn + 6ce3 f 4 kn + 10c4 d2 e2 ln − 16c2 d3 e3 ln − 14c5 def ln +58c3 d2 e2 f ln − 12cd3 e3 f ln + 4c6 f 2 ln − 68c4 def 2 ln + 48c2 d2 e2 f 2 ln +14c5 f 3 ln − 74c3 def 3 ln + 8cd2 e2 f 3 ln + 2c4 f 4 ln − 30c2 def 4 ln − 14c3 f 5 ln −6cdef 5 ln − 6c2 f 6 ln − 14c3 d2 e3 mn − 4cd3 e4 mn + 52c4 de2 f mn −88c2 d2 e3 f mn − 14c5 ef 2 mn + 204c3 de2 f 2 mn − 74cd2 e3 f 2 mn −40c4 ef 3 mn + 128c2 de2 f 3 mn + 18c3 ef 4 mn − 12cde2 f 4 mn + 36c2 ef 5 mn −6c4 de3 n2 + 31c2 d2 e4 n2 − 63c3 de3 f n2 + 27cd2 e4 f n2 + 14c4 e2 f 2 n2 −43c2 de3 f 2 n2 − 2c3 e2 f 3 n2 + 6cde3 f 3 n2 − 12c2 e2 f 4 n2 ; G1,2 = −6c3 d3 e2 g 2 − 27cd4 e3 g 2 + 18c4 d2 ef g 2 + 70c2 d3 e2 f g 2 − 31d4 e3 f g 2 −12c5 df 2 g 2 − 23c3 d2 ef 2 g 2 + 175cd3 e2 f 2 g 2 − 38c4 df 3 g 2 − 206c2 d2 ef 3 g 2 +99d3 e2 f 3 g 2 + 8c3 df 4 g 2 − 195cd2 ef 4 g 2 + 42c2 df 5 g 2 − 18d2 ef 5 g 2 +12c4 d2 e2 gh + 74c2 d3 e3 gh + 4d4 e4 gh − 36c5 def gh − 194c3 d2 e2 f gh +52cd3 e3 f gh + 24c6 f 2 gh + 106c4 def 2 gh − 378c2 d2 e2 f 2 gh − 30d3 e3 f 2 gh +86c5 f 3 gh + 524c3 def 3 gh − 178cd2 e2 f 3 gh + 14c4 f 4 gh + 518c2 def 4 gh −18d2 e2 f 4 gh − 94c3 f 5 gh + 120cdef 5 gh − 30c2 f 6 gh − 48c3 d2 e3 h2 − 8cd3 e4 h2 +144c4 de2 f h2 + 8c2 d2 e3 f h2 − 168c5 ef 2 h2 + 140c3 de2 f 2 h2 + 68cd2 e3 f 2 h2 −356c4 ef 3 h2 − 76c2 de2 f 3 h2 + 12d2 e3 f 3 h2 − 208c3 ef 4 h2 − 48cde2 f 4 h2 −36c2 ef 5 h2 + 6c5 de2 gk − 8c3 d2 e3 gk + 12cd3 e4 gk − 6c6 ef gk + 34c4 de2 f gk −44c2 d2 e3 f gk + 16d3 e4 f gk − 62c5 ef 2 gk + 70c3 de2 f 2 gk − 62cd2 e3 f 2 gk −146c4 ef 3 gk + 58c2 de2 f 3 gk − 26d2 e3 f 3 gk − 120c3 ef 4 gk + 34cde2 f 4 gk

178

Appendix 3

−44c2 ef 5 gk + 6de2 f 5 gk − 6cef 6 gk − 24c4 de3 hk + 8c2 d2 e4 hk − 4d3 e5 hk +96c5 e2 f hk − 124c3 de3 f hk + 24cd2 e4 f hk + 332c4 e2 f 2 hk − 132c2 de3 f 2 hk +16d2 e4 f 2 hk + 328c3 e2 f 3 hk − 56cde3 f 3 hk + 112c2 e2 f 4 hk − 12de3 f 4 hk +12ce2 f 5 hk − 18c5 e3 k 2 + 21c3 de4 k 2 − 5cd2 e5 k 2 − 75c4 e3 f k 2 + 43c2 de4 f k 2 −5d2 e5 f k 2 − 92c3 e3 f 2 k 2 + 28cde4 f 2 k 2 − 41c2 e3 f 3 k 2 + 6de4 f 3 k 2 − 6ce3 f 4 k 2 −6c4 d3 egl − 7c2 d4 e2 gl + 4d5 e3 gl + 6c5 d2 f gl + 2c3 d3 ef gl − 36cd4 e2 f gl +23c4 d2 f 2 gl + 81c2 d3 ef 2 gl − 29d4 e2 f 2 gl − 13c3 d2 f 3 gl + 142cd3 ef 3 gl −113c2 d2 f 4 gl + 57d3 ef 4 gl − 93cd2 f 5 gl − 18d2 f 6 gl + 12c5 d2 ehl + 14c3 d3 e2 hl −10cd4 e3 hl − 12c6 df hl + 8c4 d2 ef hl + 74c2 d3 e2 f hl − 6d4 e3 f hl − 58c5 df 2 hl −112c3 d2 ef 2 hl + 74cd3 e2 f 2 hl − 24c4 df 3 hl − 262c2 d2 ef 3 hl + 14d3 e2 f 3 hl +194c3 df 4 hl − 166cd2 ef 4 hl + 224c2 df 5 hl − 24d2 ef 5 hl + 60cdf 6 hl − 6c6 dekl −c4 d2 e2 kl − 3c2 d3 e3 kl + 6c7 f kl − 16c5 def kl − 7c3 d2 e2 f kl + 35c6 f 2 kl −10c4 def 2 kl + 3d3 e3 f 2 kl + 56c5 f 3 kl + 7cd2 e2 f 3 kl + 10c2 def 4 kl + d2 e2 f 4 kl −56c3 f 5 kl + 16cdef 5 kl − 35c2 f 6 kl + 6def 6 kl − 6cf 7 kl − 6c3 d4 el2 + 5cd5 e2 l2 +6c4 d3 f l2 − 28c2 d4 ef l2 + 5d5 e2 f l2 + 41c3 d3 f 2 l2 − 43cd4 ef 2 l2 + 92c2 d3 f 3 l2 −21d4 ef 3 l2 + 75cd3 f 4 l2 + 18d3 f 5 l2 − 18c3 d3 e2 gm − 18cd4 e3 gm+ +12c4 d2 ef gm + 2c2 d3 e2 f gm − 14d4 e3 f gm − 30c5 df 2 gm + 8c3 d2 ef 2 gm +122cd3 e2 f 2 gm − 64c4 df 3 gm − 238c2 d2 ef 3 gm + 102d3 e2 f 3 gm +130c3 df 4 gm − 186cd2 ef 4 gm + 156c2 df 5 gm + 36d2 ef 5 gm + 24c4 d2 e2 hm +28c2 d3 e3 hm − 24c5 def hm + 28c3 d2 e2 f hm + 72c6 f 2 hm − 152c4 def 2 hm −28d3 e3 f 2 hm + 240c5 f 3 hm − 28cd2 e2 f 3 hm + 152c2 def 4 hm − 24d2 e2 f 4 hm −240c3 f 5 hm + 24cdef 5 hm − 72c2 f 6 hm + 24c5 de2 km − 14c3 d2 e3 km +6cd3 e4 km − 60c6 ef km + 166c4 de2 f km − 74c2 d2 e3 f km + 10d3 e4 f km −224c5 ef 2 km + 262c3 de2 f 2 km − 74cd2 e3 f 2 km − 194c4 ef 3 km +112c2 de2 f 3 km − 14d2 e3 f 3 km + 24c3 ef 4 km − 8cde2 f 4 km + 58c2 ef 5 km −12de2 f 5 km + 12cef 6 km + 12c4 d3 elm − 16c2 d4 e2 lm + 4d5 e3 lm −12c5 d2 f lm + 56c3 d3 ef lm − 24cd4 e2 f lm − 112c4 d2 f 2 lm + 132c2 d3 ef 2 lm −8d4 e2 f 2 lm − 328c3 d2 f 3 lm + 124cd3 ef 3 lm − 332c2 d2 f 4 lm + 24d3 ef 4 lm −96cd2 f 5 lm − 12c3 d3 e2 m2 + 48c4 d2 ef m2 − 68c2 d3 e2 f m2 + 8d4 e3 f m2 +36c5 df 2 m2 + 76c3 d2 ef 2 m2 − 8cd3 e2 f 2 m2 + 208c4 df 3 m2 − 140c2 d2 ef 3 m2 +48d3 e2 f 3 m2 + 356c3 df 4 m2 − 144cd2 ef 4 m2 + 168c2 df 5 m2 + 33c4 d2 e2 gn +49c2 d3 e3 gn − 24c5 def gn − 35c3 d2 e2 f gn + 9c6 f 2 gn − 8c4 def 2 gn −49d3 e3 f 2 gn + 30c5 f 3 gn + 35cd2 e2 f 3 gn + 8c2 def 4 gn − 33d2 e2 f 4 gn −30c3 f 5 gn + 24cdef 5 gn − 9c2 f 6 gn − 36c5 de2 hn − 102c3 d2 e3 hn+ +14cd3 e4 hn + 186c4 de2 f hn − 122c2 d2 e3 f hn + 18d3 e4 f hn − 156c5 ef 2 hn +238c3 de2 f 2 hn − 2cd2 e3 f 2 hn − 130c4 ef 3 hn − 8c2 de2 f 3 hn + 18d2 e3 f 3 hn +64c3 ef 4 hn − 12cde2 f 4 hn + 30c2 ef 5 hn + 18c6 e2 kn − 57c4 de3 kn +29c2 d2 e4 kn − 4d3 e5 kn + 93c5 e2 f kn − 142c3 de3 f kn + 36cd2 e4 f kn +113c4 e2 f 2 kn − 81c2 de3 f 2 kn + 7d2 e4 f 2 kn + 13c3 e2 f 3 kn − 2cde3 f 3 kn −23c2 e2 f 4 kn + 6de3 f 4 kn − 6ce2 f 5 kn − 6c5 d2 eln + 26c3 d3 e2 ln − 16cd4 e3 ln +6c6 df ln − 34c4 d2 ef ln + 62c2 d3 e2 f ln − 12d4 e3 f ln + 44c5 df 2 ln −58c3 d2 ef 2 ln + 44cd3 e2 f 2 ln + 120c4 df 3 ln − 70c2 d2 ef 3 ln + 8d3 e2 f 3 ln +146c3 df 4 ln − 34cd2 ef 4 ln + 62c2 df 5 ln − 6d2 ef 5 ln + 6cdf 6 ln + 18c4 d2 e2 mn +30c2 d3 e3 mn − 4d4 e4 mn − 120c5 def mn + 178c3 d2 e2 f mn − 52cd3 e3 f mn +30c6 f 2 mn − 518c4 def 2 mn + 378c2 d2 e2 f 2 mn − 74d3 e3 f 2 mn + 94c5 f 3 mn

Appendix 3

179

−524c3 def 3 mn + 194cd2 e2 f 3 mn − 14c4 f 4 mn − 106c2 def 4 mn −12d2 e2 f 4 mn − 86c3 f 5 mn + 36cdef 5 mn − 24c2 f 6 mn + 18c5 de2 n2 −99c3 d2 e3 n2 + 31cd3 e4 n2 + 195c4 de2 f n2 − 175c2 d2 e3 f n2 + 27d3 e4 f n2 −42c5 ef 2 n2 + 206c3 de2 f 2 n2 − 70cd2 e3 f 2 n2 − 8c4 ef 3 n2 + 23c2 de2 f 3 n2 +6d2 e3 f 3 n2 + 38c3 ef 4 n2 − 18cde2 f 4 n2 + 12c2 ef 5 n2 ; G1,3 = −6c3 d3 ef g 2 − 27cd4 e2 f g 2 + 12c4 d2 f 2 g 2 + 43c2 d3 ef 2 g 2 −31d4 e2 f 2 g 2 + 2c3 d2 f 3 g 2 + 63cd3 ef 3 g 2 − 14c2 d2 f 4 g 2 + 6d3 ef 4 g 2 +12c4 d2 ef gh + 74c2 d3 e2 f gh + 4d4 e3 f gh − 36c5 df 2 gh − 128c3 d2 ef 2 gh +88cd3 e2 f 2 gh − 18c4 df 3 gh − 204c2 d2 ef 3 gh + 14d3 e2 f 3 gh + 40c3 df 4 gh −52cd2 ef 4 gh + 14c2 df 5 gh − 48c3 d2 e2 f h2 − 8cd3 e3 f h2 + 24c6 f 2 h2 +100c4 def 2 h2 − 48c2 d2 e2 f 2 h2 + 20c5 f 3 h2 + 148c3 def 3 h2 − 20cd2 e2 f 3 h2 −20c4 f 4 h2 + 64c2 def 4 h2 − 4d2 e2 f 4 h2 − 20c3 f 5 h2 + 8cdef 5 h2 − 4c2 f 6 h2 +6c5 def gk − 8c3 d2 e2 f gk + 12cd3 e3 f gk + 6c6 f 2 gk + 30c4 def 2 gk −48c2 d2 e2 f 2 gk + 16d3 e3 f 2 gk + 14c5 f 3 gk + 74c3 def 3 gk − 58cd2 e2 f 3 gk −2c4 f 4 gk + 68c2 def 4 gk − 10d2 e2 f 4 gk − 14c3 f 5 gk + 14cdef 5 gk −4c2 f 6 gk − 24c6 ef hk + 20c4 de2 f hk − 8c2 d2 e3 f hk − 4d3 e4 f hk −92c5 ef 2 hk + 48c3 de2 f 2 hk + 4cd2 e3 f 2 hk − 112c4 ef 3 hk + 28c2 de2 f 3 hk +4d2 e3 f 3 hk − 52c3 ef 4 hk + 4cde2 f 4 hk − 8c2 ef 5 hk + 6c6 e2 k 2 −11c4 de3 k 2 + 4c2 d2 e4 k 2 + 29c5 e2 f k 2 − 31c3 de3 f k 2 + 3cd2 e4 f k 2 +45c4 e2 f 2 k 2 − 24c2 de3 f 2 k 2 − d2 e4 f 2 k 2 + 27c3 e2 f 3 k 2 − 4cde3 f 3 k 2 +5c2 e2 f 4 k 2 − 6c4 d3 f gl − 7c2 d4 ef gl + 4d5 e2 f gl − 5c3 d3 f 2 gl −32cd4 ef 2 gl + 28c2 d3 f 3 gl − 17d4 ef 3 gl + 29cd3 f 4 gl + 6d3 f 5 gl +12c5 d2 f hl + 14c3 d3 ef hl − 10cd4 e2 f hl + 10c4 d2 f 2 hl + 80c2 d3 ef 2 hl −6d4 e2 f 2 hl − 70c3 d2 f 3 hl + 70cd3 ef 3 hl − 88c2 d2 f 4 hl + 12d3 ef 4 hl −24cd2 f 5 hl − 6c6 df kl − c4 d2 ef kl − 3c2 d3 e2 f kl − 11c5 df 2 kl −18c3 d2 ef 2 kl + 21c4 df 3 kl − 41c2 d2 ef 3 kl + 3d3 e2 f 3 kl + 53c3 df 4 kl −30cd2 ef 4 kl + 33c2 df 5 kl − 6d2 ef 5 kl + 6cdf 6 kl − 6c3 d4 f l2 + 5cd5 ef l2 −23c2 d4 f 2 l2 + 5d5 ef 2 l2 − 23cd4 f 3 l2 − 6d4 f 4 l2 + 12c5 d2 f gm −10c3 d3 ef gm − 50cd4 e2 f gm + 34c4 d2 f 2 gm + 88c2 d3 ef 2 gm −46d4 e2 f 2 gm − 38c3 d2 f 3 gm + 78cd3 ef 3 gm − 56c2 d2 f 4 gm − 12d3 ef 4 gm −24c6 df hm + 20c4 d2 ef hm + 76c2 d3 e2 f hm − 68c5 df 2 hm − 152c3 d2 ef 2 hm +96cd3 e2 f 2 hm + 76c4 df 3 hm − 212c2 d2 ef 3 hm + 20d3 e2 f 3 hm +164c3 df 4 hm − 40cd2 ef 4 hm + 44c2 df 5 hm + 12c7 f km − 22c5 def km +26c3 d2 e2 f km + 6cd3 e3 f km + 46c6 f 2 km − 64c4 def 2 km + 12c2 d2 e2 f 2 km +10d3 e3 f 2 km + 38c5 f 3 km − 8c3 def 3 km − 22cd2 e2 f 3 km − 34c4 f 4 km +50c2 def 4 km − 50c3 f 5 km + 12cdef 5 km − 12c2 f 6 km + 24c4 d3 f lm −32c2 d4 ef lm + 4d5 e2 f lm + 104c3 d3 f 2 lm − 52cd4 ef 2 lm + 120c2 d3 f 3 lm −12d4 ef 3 lm + 36cd3 f 4 lm − 24c5 d2 f m2 + 44c3 d3 ef m2 − 32cd4 e2 f m2 −116c4 d2 f 2 m2 + 132c2 d3 ef 2 m2 − 24d4 e2 f 2 m2 − 172c3 d2 f 3 m2 + +72cd3 ef 3 m2 − 72c2 d2 f 4 m2 − 6c5 d2 egn − 4c3 d3 e2 gn + 16cd4 e3 gn +13c4 d2 ef gn − 3c2 d3 e2 f gn + 16d4 e3 f gn − 21c5 df 2 gn − 6c3 d2 ef 2 gn +20cd3 e2 f 2 gn − 7c4 df 3 gn − 45c2 d2 ef 3 gn + 19d3 e2 f 3 gn + 21c3 df 4 gn −20cd2 ef 4 gn + 7c2 df 5 gn + 12c6 dehn + 2c4 d2 e2 hn − 24c2 d3 e3 hn −14c5 def hn + 18c3 d2 e2 f hn − 34cd3 e3 f hn + 36c6 f 2 hn − 24c4 def 2 hn −6d3 e3 f 2 hn + 30c5 f 3 hn + 40c3 def 3 hn − 30cd2 e2 f 3 hn − 30c4 f 4 hn

180

Appendix 3

+54c2 def 4 hn − 6d2 e2 f 4 hn − 30c3 f 5 hn + 12cdef 5 hn − 6c2 f 6 hn − 6c7 ekn +23c5 de2 kn − 20c3 d2 e3 kn − 35c6 ef kn + 76c4 de2 f kn − 23c2 d2 e3 f kn −4d3 e4 f kn − 56c5 ef 2 kn + 53c3 de2 f 2 kn + 4cd2 e3 f 2 kn − 18c4 ef 3 kn −8c2 de2 f 3 kn − d2 e3 f 3 kn + 14c3 ef 4 kn − 4cde2 f 4 kn + 5c2 ef 5 kn −6c4 d3 eln + 8c2 d4 e2 ln − 6c5 d2 f ln − 6c3 d3 ef ln − 38c4 d2 f 2 ln +34c2 d3 ef 2 ln − 4d4 e2 f 2 ln − 70c3 d2 f 3 ln + 32cd3 ef 3 ln − 40c2 d2 f 4 ln +6d3 ef 4 ln − 6cd2 f 5 ln + 12c5 d2 emn − 28c3 d3 e2 mn + 16cd4 e3 mn +12c6 df mn + 24c4 d2 ef mn + 2c2 d3 e2 f mn + 12d4 e3 f mn + 76c5 df 2 mn −144c3 d2 ef 2 mn + 52cd3 e2 f 2 mn + 198c4 df 3 mn − 200c2 d2 ef 3 mn +22d3 e2 f 3 mn + 160c3 df 4 mn − 56cd2 ef 4 mn + 34c2 df 5 mn − 6c6 den2 +35c4 d2 e2 n2 − 36c2 d3 e3 n2 − 65c5 def n2 + 129c3 d2 e2 f n2 − 41cd3 e3 f n2 +12c6 f 2 n2 − 121c4 def 2 n2 + 102c2 d2 e2 f 2 n2 − 9d3 e3 f 2 n2 + 10c5 f 3 n2 −57c3 def 3 n2 + 14cd2 e2 f 3 n2 − 10c4 f 4 n2 + 5c2 def 4 n2 − 2d2 e2 f 4 n2 −10c3 f 5 n2 + 4cdef 5 n2 − 2c2 f 6 n2 ; G1,4 = 6c3 d4 eg 2 + 27cd5 e2 g 2 − 12c4 d3 f g 2 − 43c2 d4 ef g 2 + 31d5 e2 f g 2 −2c3 d3 f 2 g 2 − 63cd4 ef 2 g 2 + 14c2 d3 f 3 g 2 − 6d4 ef 3 g 2 − 12c4 d3 egh −74c2 d4 e2 gh − 4d5 e3 gh + 36c5 d2 f gh + 128c3 d3 ef gh − 88cd4 e2 f gh +18c4 d2 f 2 gh + 204c2 d3 ef 2 gh − 14d4 e2 f 2 gh − 40c3 d2 f 3 gh + 52cd3 ef 3 gh −14c2 d2 f 4 gh + 48c3 d3 e2 h2 + 8cd4 e3 h2 − 24c6 df h2 − 100c4 d2 ef h2 +48c2 d3 e2 f h2 − 20c5 df 2 h2 − 148c3 d2 ef 2 h2 + 20cd3 e2 f 2 h2 + 20c4 df 3 h2 −64c2 d2 ef 3 h2 + 4d3 e2 f 3 h2 + 20c3 df 4 h2 − 8cd2 ef 4 h2 + 4c2 df 5 h2 +12c3 d3 e2 gk − 28cd4 e3 gk − 12c6 df gk − 14c4 d2 ef gk + 116c2 d3 e2 f gk −32d4 e3 f gk − 34c5 df 2 gk − 126c3 d2 ef 2 gk + 134cd3 e2 f 2 gk + 2c4 df 3 gk −148c2 d2 ef 3 gk + 22d3 e2 f 3 gk + 34c3 df 4 gk − 32cd2 ef 4 gk + 10c2 df 5 gk +12c6 dehk − 22c4 d2 e2 hk + 32c2 d3 e3 hk + 4d4 e4 hk + 12c7 f hk +48c5 def hk − 148c3 d2 e2 f hk + 28cd3 e3 f hk + 46c6 f 2 hk + 168c4 def 2 hk −180c2 d2 e2 f 2 hk + 4d3 e3 f 2 hk + 20c5 f 3 hk + 212c3 def 3 hk − 64cd2 e2 f 3 hk −40c4 f 4 hk + 92c2 def 4 hk − 6d2 e2 f 4 hk − 32c3 f 5 hk + 12cdef 5 hk −6c2 f 6 hk − 6c7 ek 2 + 17c5 de2 k 2 − 9c3 d2 e3 k 2 + 5cd3 e4 k 2 − 35c6 ef k 2 +59c4 de2 f k 2 − 26c2 d2 e3 f k 2 + 5d3 e4 f k 2 − 68c5 ef 2 k 2 + 73c3 de2 f 2 k 2 −20cd2 e3 f 2 k 2 − 58c4 ef 3 k 2 + 37c2 de2 f 3 k 2 − 3d2 e3 f 3 k 2 − 22c3 ef 4 k 2 +6cde2 f 4 k 2 − 3c2 ef 5 k 2 + 6c4 d4 gl + 7c2 d5 egl − 4d6 e2 gl + 5c3 d4 f gl +32cd5 ef gl − 28c2 d4 f 2 gl + 17d5 ef 2 gl − 29cd4 f 3 gl − 6d4 f 4 gl − 12c5 d3 hl −14c3 d4 ehl + 10cd5 e2 hl − 10c4 d3 f hl − 80c2 d4 ef hl + 6d5 e2 f hl +70c3 d3 f 2 hl − 70cd4 ef 2 hl + 88c2 d3 f 3 hl − 12d4 ef 3 hl + 24cd3 f 4 hl +6c6 d2 kl + 7c4 d3 ekl − 5c2 d4 e2 kl + 5c5 d2 f kl + 58c3 d3 ef kl − 16cd4 e2 f kl −53c4 d2 f 2 kl + 109c2 d3 ef 2 kl − 11d4 e2 f 2 kl − 105c3 d2 f 3 kl + 70cd3 ef 3 kl −65c2 d2 f 4 kl + 12d3 ef 4 kl − 12cd2 f 5 kl + 6c3 d5 l2 − 5cd6 el2 + 23c2 d5 f l2 −5d6 ef l2 + 23cd5 f 2 l2 + 6d5 f 3 l2 − 12c5 d3 gm + 10c3 d4 egm + 50cd5 e2 gm −34c4 d3 f gm − 88c2 d4 ef gm + 46d5 e2 f gm + 38c3 d3 f 2 gm − 78cd4 ef 2 gm +56c2 d3 f 3 gm + 12d4 ef 3 gm + 24c6 d2 hm − 20c4 d3 ehm − 76c2 d4 e2 hm +68c5 d2 f hm + 152c3 d3 ef hm − 96cd4 e2 f hm − 76c4 d2 f 2 hm +212c2 d3 ef 2 hm − 20d4 e2 f 2 hm − 164c3 d2 f 3 hm + 40cd3 ef 3 hm −44c2 d2 f 4 hm − 12c7 dkm + 10c5 d2 ekm + 2c3 d3 e2 km − 22cd4 e3 km −34c6 df km − 40c4 d2 ef km + 104c2 d3 e2 f km − 26d4 e3 f km + 38c5 df 2 km

Appendix 3

181

−248c3 d2 ef 2 km + 122cd3 e2 f 2 km + 190c4 df 3 km − 250c2 d2 ef 3 km +12d3 e2 f 3 km + 166c3 df 4 km − 48cd2 ef 4 km + 36c2 df 5 km − 24c4 d4 lm +32c2 d5 elm − 4d6 e2 lm − 104c3 d4 f lm + 52cd5 ef lm − 120c2 d4 f 2 lm +12d5 ef 2 lm − 36cd4 f 3 lm + 24c5 d3 m2 − 44c3 d4 em2 + 32cd5 e2 m2 +116c4 d3 f m2 − 132c2 d4 ef m2 + 24d5 e2 f m2 + 172c3 d3 f 2 m2 −72cd4 ef 2 m2 + 72c2 d3 f 3 m2 + 6c6 d2 gn − 29c4 d3 egn − 65c2 d4 e2 gn +41c5 d2 f gn + 58c3 d3 ef gn − 96cd4 e2 f gn + 7c4 d2 f 2 gn + 125c2 d3 ef 2 gn −31d4 e2 f 2 gn − 41c3 d2 f 3 gn + 38cd3 ef 3 gn − 13c2 d2 f 4 gn − 12c7 dhn +58c5 d2 ehn + 82c3 d3 e2 hn + 2cd4 e3 hn − 82c6 df hn − 32c4 d2 ef hn +152c2 d3 e2 f hn − 2d4 e3 f hn − 50c5 df 2 hn − 200c3 d2 ef 2 hn +90cd3 e2 f 2 hn + 70c4 df 3 hn − 138c2 d2 ef 3 hn + 12d3 e2 f 3 hn + 62c3 df 4 hn −24cd2 ef 4 hn + 12c2 df 5 hn + 6c8 kn − 29c6 dekn + 31c4 d2 e2 kn +11c2 d3 e3 kn + 4d4 e4 kn + 41c7 f kn − 110c5 def kn − 7c3 d2 e2 f kn +16cd3 e3 f kn + 79c6 f 2 kn − 26c4 def 2 kn − 90c2 d2 e2 f 2 kn + 13d3 e3 f 2 kn +10c5 f 3 kn + 140c3 def 3 kn − 65cd2 e2 f 3 kn − 76c4 f 4 kn + 103c2 def 4 kn −9d2 e2 f 4 kn − 51c3 f 5 kn + 18cdef 5 kn − 9c2 f 6 kn + 12c5 d3 ln −34c3 d4 eln + 16cd5 e2 ln + 70c4 d3 f ln − 102c2 d4 ef ln + 12d5 e2 f ln +122c3 d3 f 2 ln − 72cd4 ef 2 ln + 72c2 d3 f 3 ln − 12d4 ef 3 ln + 12cd3 f 4 ln −24c6 d2 mn + 80c4 d3 emn − 118c2 d4 e2 mn + 4d5 e3 mn − 152c5 d2 f mn +400c3 d3 ef mn − 152cd4 e2 f mn − 354c4 d2 f 2 mn + 400c2 d3 ef 2 mn −34d4 e2 f 2 mn − 276c3 d2 f 3 mn + 92cd3 ef 3 mn − 58c2 d2 f 4 mn + 6c7 dn2 −29c5 d2 en2 + 91c3 d3 e2 n2 − 15cd4 e3 n2 + 47c6 df n2 − 247c4 d2 ef n2 +200c2 d3 e2 f n2 − 11d4 e3 f n2 + 174c5 df 2 n2 − 403c3 d2 ef 2 n2 +118cd3 e2 f 2 n2 + 224c4 df 3 n2 − 215c2 d2 ef 3 n2 + 17d3 e2 f 3 n2 +108c3 df 4 n2 − 34cd2 ef 4 n2 + 17c2 df 5 n2 ; σ1,0 = 12c4 d2 e4 − 22c2 d3 e5 + 8d4 e6 − 24c5 de3 f + 114c3 d2 e4 f − 76cd3 e5 f +12c6 e2 f 2 − 162c4 de3 f 2 + 252c2 d2 e4 f 2 − 22d3 e5 f 2 + 70c5 e2 f 3 −308c3 de3 f 3 + 114cd2 e4 f 3 + 124c4 e2 f 4 − 162c2 de3 f 4 + 12d2 e4 f 4 +70c3 e2 f 5 − 24cde3 f 5 + 12c2 e2 f 6 ; σ1,1 = 6c5 d2 e3 + 11c3 d3 e4 − 4cd4 e5 + 12c6 de2 f − 51c4 d2 e3 f + 27c2 d3 e4 f +4d4 e5 f − 6c7 ef 2 + 69c5 de2 f 2 − 69c3 d2 e3 f 2 − 27cd3 e4 f 2 − 29c6 ef 3 +73c4 de2 f 3 + 69c2 d2 e3 f 3 − 11d3 e4 f 3 − 27c5 ef 4 − 73c3 de2 f 4 +51cd2 e3 f 4 + 27c4 ef 5 − 69c2 de2 f 5 + 6d2 e3 f 5 + 29c3 ef 6 −12cde2 f 6 + 6c2 ef 7 ; σ1,2 = 6c6 d2 e2 − 23c4 d3 e3 + 26c2 d4 e4 − 8d5 e5 − 12c7 def + 69c5 d2 e2 f −130c3 d3 e3 f + 68cd4 e4 f + 6c8 f 2 − 69c6 def 2 + 180c4 d2 e2 f 2 −198c2 d3 e3 f 2 + 26d4 e4 f 2 + 23c7 f 3 − 74c5 def 3 + 170c3 d2 e2 f 3 −130cd3 e3 f 3 − 2c6 f 4 + 22c4 def 4 + 180c2 d2 e2 f 4 − 23d3 e3 f 4 − 54c5 f 5 −74c3 def 5 + 69cd2 e2 f 5 − 2c4 f 6 − 69c2 def 6 + 6d2 e2 f 6 + 23c3 f 7 −12cdef 7 + 6c2 f 8 ; σ1,3 = 6c5 d3 e2 − 11c3 d4 e3 + 4cd5 e4 − 12c6 d2 ef + 51c4 d3 e2 f − 27c2 d4 e3 f −4d5 e4 f + 6c7 df 2 − 69c5 d2 ef 2 + 69c3 d3 e2 f 2 + 27cd4 e3 f 2 + 29c6 df 3 −73c4 d2 ef 3 − 69c2 d3 e2 f 3 + 11d4 e3 f 3 + 27c5 df 4 + 73c3 d2 ef 4 −51cd3 e2 f 4 − 27c4 df 5 + 69c2 d2 ef 5 − 6d3 e2 f 5 − 29c3 df 6 + 12cd2 ef 6 −6c2 df 7 ;

182

Appendix 3 σ1,4 = 12c4 d4 e2 − 22c2 d5 e3 + 8d6 e4 − 24c5 d3 ef + 114c3 d4 e2 f −76cd5 e3 f + 12c6 d2 f 2 − 162c4 d3 ef 2 + 252c2 d4 e2 f 2 − 22d5 e3 f 2 +70c5 d2 f 3 − 308c3 d3 ef 3 + 114cd4 e2 f 3 + 124c4 d2 f 4 − 162c2 d3 ef 4 +12d4 e2 f 4 + 70c3 d2 f 5 − 24cd3 ef 5 + 12c2 d2 f 6 ;

B1,i = 4ki (c + f )(cf − de)2 (2c2 − de + 5cf + 2f 2 )(3c2 − 4de + 10cf + 3f 2 ), (k0 = k1 = k3 = k4 = 1, k2 = 3). (.38)

Appendix 4

Matrices that Define a Linear System of Equations A2 B2 = C2 for the Quantity G2 in Case of the Differential Systems(1, 2)  A2 = [A2 |A2 |A 2 |A2 ],

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  ⎜ A2 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

3c 3d 0 0 3g 6h 3k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3e 3(2c + f ) 6d 0 3l 6(g + m) 3(4h + n) 6k 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 6e 0 3(2c + f ) 3e 3d 3f 0 0 6l 0 3(g + 4m) 3l 6(h + n) 6m 3k 3n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 4c 4d 0 0 0 4g 8h 4k 0 0 0 0 0 0 0 0 0 0

0 0 0 0 4e 4(f + 3c) 12d 0 0 4l 4(3g + 2m) 4(6h + n) 12k 0 0 0 0 0 0 0 0 0

0 0 0 0 0 12e 12(c + f ) 12d 0 0 12l 12(g + 2m) 12(2h + n) 12k 0 0 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

183

184 ⎛

0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 12e ⎜ ⎜ 4(3f + c) ⎜ ⎜ 4d ⎜ ⎜ 0 ⎜ ⎜ 0  A2 = ⎜ ⎜ 12l ⎜ ⎜ 4(g + 6m) ⎜ ⎜ 4(2h + 3n) ⎜ ⎜ 4k ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0 ⎛

A 2

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 0 0 0 0 −e2 0 2e(c − f ) 0 2de − (c − f )2 4l 2d(f − c) 4f − d2 0 0 0 0 0 0 4l 0 8m 0 4n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 30e 10(2c + 3f ) 20d 0 0 0 30l 20(g + 3m) 10(4h + 3n) 20k 0

0 0 0 0 0 0 0 0 0 0 0 0 20e 5(c + 4f ) 5d 0 0 0 20l 5(g + 8m) 10(h + 2n) 5k

0 0 0 0 0 0 0 0 0 5c 5d 0 0 0 0 5g 10h 5k 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5e 0 5f 0 0 6c 0 6d 0 0 0 0 5l 0 10m 0 5n 0

0 0 0 0 0 0 0 0 0 5e 5(4c + f ) 20d 0 0 0 5l 10(2g + m) 5(8h + n) 20k 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6e 6(5c + f ) 30d 0 0 0 0

Appendix 4 ⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 20e ⎟, 10(3c + 2f ) ⎟ ⎟ ⎟ 30d ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 20l ⎟ 10(3g + 4m) ⎟ ⎟ 20(3h + n) ⎟ ⎟ ⎟ 30k ⎟ ⎠ 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30e 30(2c + f ) 60d 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Appendix 4 ⎛

A 2

0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ =⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 60e ⎜ ⎜ 60(c + f ) ⎜ ⎜ 60d ⎜ ⎝ 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60e 30(c + 2f ) 30d 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30e 6(c + 5f ) 6d

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6e 6f

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e3 3e2 (f − c) 3e[(c − f )2 − de] (f − c)[(c − f )2 − 6de] 3d[de − (c − f )2 ] 3d2 (f − c) −d3

185 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

186

Appendix 4 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ B2 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

a0 a1 a2 a3 b0 b1 b2 b3 b4 G1 c0 c1 c2 c3 c4 c5 d0 d1 d2 d3 d4 d5 d6 G2

⎞

⎛ ⎟ 2eg + (f − c)l ⎟ ⎟ ⎜ (f − c)(g + 2m) − 2dl + 4eh ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (f − c)(2h + n) + 2ek − 4dm ⎟ ⎜ (f − c)k − 2dn ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎟ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎜ ⎟ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ , C2 = ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎝ 0 ⎟ ⎠ 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Appendix 5

Polynomials that Define the Quantity G1 for the System s(1, 3) Expressions of Focus Pseudo-Quantities G1,i , and σ1,i , B1,i (i = 0, 1, 2, 3, 4) G1,0 = 7c2 de2 p − 6d2 e3 p − 7c3 ef p + 35cde2 f p − 29c2 ef 2 p + 22de2 f 2 p −25cef 3 p − 3ef 4 p − 9cde3 q + 9c2 e2 f q − 21de3 f q + 30ce2 f 2 q + 9e2 f 3 q +6de4 r − 15ce3 f r − 9e3 f 2 r + 3ce4 s + 3e4 f s − 3c3 det + 5cd2 e2 t + 3c4 f t −15c2 def t + 5d2 e2 f t + 10c3 f 2 t − 5cdef 2 t + 7def 3 t − 10cf 4 t − 3f 5 t +3c2 de2 u − 6d2 e3 u − 3c3 ef u + 15cde2 f u − 9c2 ef 2 u + 6de2 f 2 u + 3cef 3 u +9ef 4 u − 3cde3 v + 3c2 e2 f v − 15de3 f v + 6ce2 f 2 v − 9e2 f 3 v + 6de4 w −3ce3 f w + 3e3 f 2 w; G1,1 = −c3 dep − 2cd2 e2 p + c4 f p − c2 def p − 8d2 e2 f p + 3c3 f 2 p +12cdef 2 p − c2 f 3 p + 6def 3 p − 3cf 4 p + 9c2 de2 q − 9c3 ef q + 21cde2 f q −30c2 ef 2 q − 9cef 3 q − 6cde3 r + 15c2 e2 f r + 9ce2 f 2 r − 3c2 e3 s − 3ce3 f s −c2 d2 et + c3 df t − 5cd2 ef t + 5c2 df 2 t − 4d2 ef 2 t + 7cdf 3 t + 3df 4 t −3c3 deu + 6cd2 e2 u + 3c4 f u − 15c2 def u + 9c3 f 2 u − 6cdef 2 u − 3c2 f 3 u −9cf 4 u + 3c2 de2 v − 3c3 ef v + 15cde2 f v − 6c2 ef 2 v + 9cef 3 v − 6cde3 w +3c2 e2 f w − 3ce2 f 2 w; G1,2 = 3(c2 d2 ep + 2d3 e2 p − c3 df p − 3cd2 ef p − 2c2 df 2 p − 6d2 ef 2 p +3cdf 3 p − 3c3 deq − 5cd2 e2 q + 3c4 f q + 7c2 def q − d2 e2 f q + 7c3 f 2 q +17cdef 2 q − 7c2 f 3 q + 3def 3 q − 3cf 4 q + 6c2 de2 r − 2d2 e3 r − 15c3 ef r +7cde2 f r − 14c2 ef 2 r + 3de2 f 2 r − 3cef 3 r + 3c3 e2 s − cde3 s + 4c2 e2 f s −de3 f s + ce2 f 2 s + cd3 et − c2 d2 f t + d3 ef t − 4cd2 f 2 t − 3d2 f 3 t −3c2 d2 eu + 2d3 e2 u + 3c3 df u − 7cd2 ef u + 14c2 df 2 u − 6d2 ef 2 u +15cdf 3 u − 3c3 dev + cd2 e2 v + 3c4 f v − 17c2 def v + 5d2 e2 f v + 7c3 f 2 v −7cdef 2 v − 7c2 f 3 v + 3def 3 v − 3cf 4 v + 6c2 de2 w − 2d2 e3 w − 3c3 ef w +3cde2 f w + 2c2 ef 2 w − de2 f 2 w + cef 3 w); G1,3 = 3c2 d2 f p + 6d3 ef p − 3cd2 f 2 p − 9c3 df q − 15cd2 ef q + 6c2 df 2 q −3d2 ef 2 q + 3cdf 3 q + 9c4 f r + 6c2 def r − 6d2 e2 f r + 3c3 f 2 r + 15cdef 2 r −9c2 f 3 r + 3def 3 r − 3cf 4 r − 3c4 es + 4c2 de2 s − 7c3 ef s + 5cde2 f s −5c2 ef 2 s + de2 f 2 s − cef 3 s + 3cd3 f t + 3d3 f 2 t − 9c2 d2 f u + 6d3 ef u −15cd2 f 2 u + 9c3 df v − 21cd2 ef v + 30c2 df 2 v − 9d2 ef 2 v + 9cdf 3 v −6c3 dew + 8cd2 e2 w + 3c4 f w − 12c2 def w + 2d2 e2 f w + c3 f 2 w +cdef 2 w − 3c2 f 3 w + def 3 w − cf 4 w; G1,4 = −(3c2 d3 p + 6d4 ep − 3cd3 f p − 9c3 d2 q − 15cd3 eq + 6c2 d2 f q −3d3 ef q + 3cd2 f 2 q + 9c4 dr + 6c2 d2 er − 6d3 e2 r + 3c3 df r + 15cd2 ef r 187

188

Appendix 5

−9c2 df 2 r + 3d2 ef 2 r − 3cdf 3 r − 3c5 s + 7c3 des + 5cd2 e2 s − 10c4 f s −5c2 def s + 5d2 e2 f s − 15cdef 2 s + 10c2 f 3 s − 3def 3 s + 3cf 4 s + 3cd4 t +3d4 f t − 9c2 d3 u + 6d4 eu − 15cd3 f u + 9c3 d2 v − 21cd3 ev + 30c2 d2 f v −9d3 ef v + 9cd2 f 2 v − 3c4 dw + 22c2 d2 ew − 6d3 e2 w − 25c3 df w + 35cd2 ef w −29c2 df 2 w + 7d2 ef 2 w − 7cdf 3 w); σ1,0 = −6c2 de3 + 8d2 e4 + 6c3 e2 f − 28cde3 f + 20c2 e2 f 2 − 6de3 f 2 + 6ce2 f 3 ; σ1,1 = 3c3 de2 − 4cd2 e3 − 3c4 ef + 11c2 de2 f + 4d2 e3 f − 7c3 ef 2 − 11cde2 f 2 +7c2 ef 3 − 3de2 f 3 + 3cef 4 ; σ1,2 = −3c4 de + 10c2 d2 e2 − 8d3 e3 + 3c5 f − 14c3 def + 20cd2 e2 f + 4c4 f 2 +2c2 def 2 + 10d2 e2 f 2 − 14c3 f 3 − 14cdef 3 + 4c2 f 4 − 3def 4 + 3cf 5 ; σ1,3 = −3c3 d2 e + 4cd3 e2 + 3c4 df − 11c2 d2 ef − 4d3 e2 f + 7c3 df 2 + 11cd2 ef 2 −7c2 df 3 + 3d2 ef 3 − 3cdf 4 ; σ1,4 = −6c2 d3 e + 8d4 e2 + 6c3 d2 f − 28cd3 ef + 20c2 d2 f 2 − 6d3 ef 2 + 6cd2 f 3 ; B1,i = 4ki (c + f )(−de + cf )(3c2 − 4de + 10cf + 3f 2 ), (k0 = k1 = k3 = k4 = 1, k2 = 3).

Appendix 6

Matrices that Define a Linear System of Equations A2 B2 = C2 for the Quantity G2 in Case of the Differential System s(1, 3) A2 = [A2 |A2 |A 2 ],

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  A2 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ A2 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

4c 4d 0 0 0 4p 12q 12r 4s 0 0 0

4e 0 12c + 4f 12e 12d 12c + 12f 0 12d 0 0 4t 0 12p + 12u 12t 36q + 12v 12p + 36u 36r + 4w 36q + 36v 12s 36r + 12w 0 12s 0 0

−e2 2ce − 2ef −c2 + 2de + 2cf − f 2 −2cd + 2df −d2 0 0 0 0 0 0 0

0 0 0 0 0 6c 6d 0 0 0 0 0

0 0 12e 4c + 12f 4d 0 0 12t 4p + 36u 12q + 36v 12r + 12w 4s

0 0 0 0 0 6e 30c + 6f 30d 0 0 0 0

0 0 0 4e 4f 0 0 0 4t 12u 12v 4w

0 0 0 0 0 0 30e 60c + 30f 60d 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 0 0 0 0 0 0 60e

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 60d ⎟ 0 ⎠ 0 189

190 ⎛

A 2

Appendix 6 ⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟ e3 ⎟ , −3e2 (c − f ) ⎟ ⎟ 2 3e[(c − f ) − de] ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ −3d2 (c − f ) ⎠ −d3

0 0 0 ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎜ ⎜ 0 0 0 =⎜ ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎜ ⎜ − f )2 − 6de] 60e 0 −(c − f )[(c ⎜ ⎜ 30e −3d[(c − f )2 − de] ⎜ 30c + 60f ⎝ 30d 6c + 30f 6e 0 6d 6f ⎛ ⎞ b0 ⎛ ⎜ b1 ⎟ 2ep − ct + f t ⎟ ⎜ ⎜ b2 ⎟ ⎜ −cp + f p + 6eq − 2dt − 3cu + 3f u ⎜ ⎟ ⎜ ⎜ b3 ⎟ ⎜ −3cq + 3f q + 6er − 6du − 3cv + 3f v ⎜ ⎟ ⎜ ⎜ b4 ⎟ ⎜ ⎟ ⎜ −3cr + 3f r + 2es − 6dv − cw + f w ⎜ ⎜ G1 ⎟ ⎜ −cs + f s − 2dw ⎟ ⎜ ⎜ ⎜ d0 ⎟ ⎜ 0 ⎜ ⎟ , C2 = ⎜ B2 = ⎜ ⎜ ⎟ 0 d 1 ⎟ ⎜ ⎜ ⎜ d2 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ d3 ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎜ d4 ⎟ ⎜ 0 ⎟ ⎜ ⎜ ⎜ d5 ⎟ ⎝ 0 ⎜ ⎟ ⎝ d6 ⎠ 0 G2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Appendix 7

Matrices that Define a Linear System of Equations A3 B3 = C3 for the Quantity G3 in Case of the Differential System s(1, 4) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ A3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

 A3 = [A3 |A3 |A 3 |A3 ],

5c 5d 0 0 0 0 5g 20h 30i 20j 5k 0 0 0 0 ⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜  A3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

5e 20c + 5f 20d 0 0 0 5l 20g + 20m 80h + 30n 120i + 20o 80j + 5p 20k 0 0 0 0 0 0 0 0 0 8c 8d 0 0 0 0 0 0 0

0 0 0 0 0 0 8e 56c + 8f 56d 0 0 0 0 0 0

0 20e 30c + 20f 30d 0 0 0 20l 30g + 80m 120h + 120n 180i + 80o 120j + 20p 30k 0 0

0 0 30e 20c + 30f 20d 0 0 0 30l 20g + 120m 80h + 180n 120i + 120o 80j + 30p 20k 0

0 0 0 20e 5c + 20f 5d 0 0 0 20l 5g + 80m 20h + 120n 30i + 80o 20j + 20p 5k

0 0 0 0 0 0 0 56e 168c + 56f 168d 0 0 0 0 0

0 0 0 0 0 0 0 0 168e 280c + 168f 280d 0 0 0 0

0 0 0 0 0 0 0 0 0 280e 280c + 280f 280d 0 0 0

0 0 0 0 5e 5f 0 0 0 0 5l 20m 30n 20o 5p

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

191

192

Appendix 7 ⎛

A 3

⎛

A 3

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 0 =⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎜ 280e ⎜ ⎜ 168c + 280f ⎜ ⎜ 168d ⎜ ⎝ 0 0

0 0 0 0 0 0 0 0 0 0 0 168e 56c + 168f 56d 0

0 0 0 0 0 0 0 0 0 0 0 0 56e 8c + 56f 8d

0 0 0 0 0 0 0 0 0 0 0 0 0 8e 8f

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 0 0 0 0 0 −e4 4ce3 − 4e3 f 2 2 −6c e + 4de3 + 12ce2 f − 6e2 f 2 3 4c e − 12cde2 − 12c2 ef + 12de2 f + 12cef 2 − 4ef 3 4 −c + 12c2 de − 6d2 e2 + 4c3 f − 24cdef − 6c2 f 2 + 12def 2 +4cf 3 − f 4 3 2 −4c d + 12cd e + 12c2 df − 12d2 ef − 12cdf 2 + 4df 3 −6c2 d2 + 4d3 e + 12cd2 f − 6d2 f 2 −4cd3 + 4d3 f −d4

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Appendix 7 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ B3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

193 c0 c1 c2 c3 c4 c5 f0 f1 f2 f3 f4 f5 f6 f7 f8 G3

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ C3 = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

2eg − cl + f l −cg + f g + 8eh − 2dl − 4cm + 4f m −4ch + 4f h + 12ei − 8dm − 6cn + 6f n −6ci + 6f i + 8ej − 12dn − 4co + 4f o −4cj + 4f j + 2ek − 8do − cp + f p −ck + f k − 2dp 0 0 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Appendix 8

Matrices that Define a Linear System of Equations A2 B2 = C2 for the Quantity G2 in Case of Differential System s(1, 5) A2 = [A2 |A2 ], ⎛

6c 6d 0 0 0 0 0

6e 0 0 0 0 ⎜ 30c + 6f 30e 0 0 0 ⎜ ⎜ 30d 60c + 30f 60e 0 0 ⎜ 0 60d 60c + 60f 60e 0 A2 = ⎜ ⎜ ⎜ 0 0 60d 30c + 60f 30e ⎜ ⎝ 0 0 0 30d 6c + 30f 0 0 0 0 6d ⎛ ⎞ 0 e3 ⎟ ⎜ 0 −3ce2 + 3e2 f ⎟ ⎜ 2 2 2 ⎟ ⎜ 0 3c e − 3de − 6cef + 3ef ⎟ ⎜ ⎜ 0 −c3 + 6cde + 3c2 f − 6def − 3cf 2 + f 3 ⎟ , A2 = ⎜ ⎟ ⎜ 0 ⎟ −3c2 d + 3d2 e + 6cdf − 3df 2 ⎟ ⎜ 2 2 ⎝ 6e ⎠ −3cd + 3d f 6f −d3 ⎛ ⎞ ⎛ d0 2eg − cp + f p ⎜ d1 ⎟ ⎟ ⎜ ⎜ cg + f g + 10eh − 2dp − 5cq + 5f q − ⎜ d2 ⎟ ⎜ ⎜ ⎟ ⎜ −5ch + 5f h + 20ek − 10dq − 10cr + 10f r ⎜ d3 ⎟ ⎜ ⎟ , C2 = ⎜ −10ck + 10f k + 20el − 20dr − 10cs + 10f s ⎜ B2 = ⎜ ⎟ ⎜ d ⎜ ⎜ 4 ⎟ ⎜ d5 ⎟ ⎜ −10cl + 10f l + 10em − 20ds − 5cu + 5f u ⎟ ⎜ ⎝ −5cm + 5f m + 2en − 10du − cv + f v ⎝ d6 ⎠ −cn + f n − 2dv G2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠

195

Appendix 9

Polynomials that Define the Quantity G2 for the Differential System s(1, 5) Expressions of Focus Pseudo-Quantities G2,i and σ2,i , D2,i (i = 0, 1, 2, 3, 4, 5, 6). G20 = 22c4 de2 g − 76c2 d2 e3 g + 20d3 e4 g − 22c5 ef g + 265c3 de2 f g − 282cd2 e3 f g − 189c4 ef 2 g + 741c2 de2 f 2 g − 186d2 e3 f 2 g − 479c3 ef 3 g + 635cde2 f 3 g − 449c2 ef 4 g + 137de2 f 4 g − 147cef 5 g − 10ef 6 g − 25c3 de3 h + 70cd2 e4 h + 25c4 e2 f h − 275c2 de3 f h + 150d2 e4 f h + 205c3 e2 f 2 h − 615cde3 f 2 h + 465c2 e2 f 3 h − 285de3 f 3 h + 335ce2 f 4 h + 50e2 f 5 h+ + 30c2 de4 k − 40d2 e5 k − 30c3 e3 f k + 260cde4 f k − 220c2 e3 f 2 k + 270de4 f 2 k − 370ce3 f 3 k − 100e3 f 4 k − 40cde5 l + 40c2 e4 f l − 120de5 f l + 220ce4 f 2 l + 100e4 f 3 l + 20de6 m − 70ce5 f m − 50e5 f 2 m + 10ce6 n + 10e6 f n − 10c5 dep + 39c3 d2 e2 p − 22cd3 e3 p + 10c6 f p − 116c4 def p + 147c2 d2 e2 f p − 22d3 e3 f p + 77c5 f 2 p − 265c3 def 2 p + 81cd2 e2 f 2 p + 140c4 f 3 p − 59c2 def 3 p − 27d2 e2 f 3 p + 167cdef 4 p − 140c2 f 5 p + 67def 5 p − 77cf 6 p − 10f 7 p + 10c4 de2 q − 40c2 d2 e3 q + 20d3 e4 q − 10c5 ef q + 115c3 de2 f q − 150cd2 e3 f q − 75c4 ef 2 q + 255c2 de2 f 2 q − 90d2 e3 f 2 q − 125c3 ef 3 q + 65cde2 f 3 q + 25c2 ef 4 q − 85de2 f 4 q + 135cef 5 q + 50ef 6 q − 10c3 de3 r + 40cd2 e4 r + 10c4 e2 f r − 110c2 de3 f r + 120d2 e4 f r + 70c3 e2 f 2 r − 210cde3 f 2 r + 90c2 e2 f 3 r − 30de3 f 3 r − 70ce2 f 4 r − 100e2 f 5 r + 10c2 de4 s − 40d2 e5 s − 10c3 e3 f s + 100cde4 f s − 60c2 e3 f 2 s + 130de4 f 2 s − 30ce3 f 3 s + 100e3 f 4 s − 10cde5 u + 10c2 e4 f u − 90de5 f u + 40ce4 f 2 u − 50e4 f 3 u + 20de6 v − 10ce5 f v + 10e5 f 2 v; G21 = 2c5 deg − 2c3 d2 e2 g − 12cd3 e3 g − 2c6 f g + 17c4 def g + 36c2 d2 e2 f g − 32d3 e3 f g − 15c5 f 2 g + c3 def 2 g + 132cd2 e2 f 2 g − 25c4 f 3 g − 105c2 def 3 g + 74d2 e2 f 3 g + 5c3 f 4 g − 111cdef 4 g + 27c2 f 5 g − 20def 5 g + 10cf 6 g − 25c4 de2 h + 70c2 d2 e3 h + 25c5 ef h − 275c3 de2 f h + 150cd2 e3 f h + 205c4 ef 2 h − 615c2 de2 f 2 h + 465c3 ef 3 h − 285cde2 f 3 h 197

198

Appendix 9 + 335c2 ef 4 h + 50cef 5 h + 30c3 de3 k − 40cd2 e4 k − 30c4 e2 f k + 260c2 de3 f k − 220c3 e2 f 2 k + 270cde3 f 2 k − 370c2 e2 f 3 k − 100ce2 f 4 k − 40c2 de4 l + 40c3 e3 f l − 120cde4 f l + 220c2 e3 f 2 l + 100ce3 f 3 l + 20cde5 m − 70c2 e4 f m − 50ce4 f 2 m + 10c2 e5 n + 10ce5 f n + 2c4 d2 ep − 6c2 d3 e2 p − 2c5 df p + 25c3 d2 ef p − 22cd3 e2 f p − 19c4 df 2 p + 81c2 d2 ef 2 p − 16d3 e2 f 2 p − 59c3 df 3 p + 95cd2 ef 3 p − 79c2 df 4 p + 37d2 ef 4 p − 47cdf 5 p − 10df 6 p + 10c5 deq − 40c3 d2 e2 q + 20cd3 e3 q − 10c6 f q + 115c4 def q − 150c2 d2 e2 f q − 75c5 f 2 q + 255c3 def 2 q − 90cd2 e2 f 2 q − 125c4 f 3 q + 65c2 def 3 q + 25c3 f 4 q − 85cdef 4 q + 135c2 f 5 q + 50cf 6 q − 10c4 de2 r + 40c2 d2 e3 r + 10c5 ef r − 110c3 de2 f r + 120cd2 e3 f r + 70c4 ef 2 r − 210c2 de2 f 2 r + 90c3 ef 3 r − 30cde2 f 3 r − 70c2 ef 4 r − 100cef 5 r + 10c3 de3 s − 40cd2 e4 s − 10c4 e2 f s + 100c2 de3 f s − 60c3 e2 f 2 s + 130cde3 f 2 s − 30c2 e2 f 3 s + 100ce2 f 4 s − 10c2 de4 u + 10c3 e3 f u − 90cde4 f u + 40c2 e3 f 2 u − 50ce3 f 3 u + 20cde5 v − 10c2 e4 f v + 10ce4 f 2 v;

G22 = 5(−c4 d2 eg + 4d4 e3 g + c5 df g − 7c3 d2 ef g − 22cd3 e2 f g + 7c4 df 2 g + 9c2 d2 ef 2 g − 34d3 e2 f 2 g + 9c3 df 3 g + 51cd2 ef 3 g − 7c2 df 4 g + 20d2 ef 4 g − 10cdf 5 g + 5c5 deh − c3 d2 e2 h − 18cd3 e3 h − 5c6 f h + 37c4 def h + 99c2 d2 e2 f h − 2d3 e3 f h − 36c5 f 2 h − 29c3 def 2 h + 165cd2 e2 f 2 h − 52c4 f 3 h − 239c2 def 3 h + 17d2 e2 f 3 h + 26c3 f 4 h − 124cdef 4 h + 57c2 f 5 h − 10def 5 h + 10cf 6 h − 30c4 de2 k + 46c2 d2 e3 k − 8d3 e4 k + 30c5 ef k − 272c3 de2 f k + 60cd2 e3 f k + 226c4 ef 2 k − 366c2 de2 f 2 k + 54d2 e3 f 2 k + 414c3 ef 3 k − 128cde2 f 3 k + 174c2 ef 4 k − 20de2 f 4 k + 20cef 5 k + 40c3 de3 l − 8cd2 e4 l − 40c4 e2 f l + 136c2 de3 f l − 24d2 e4 f l − 228c3 e2 f 2 l + 68cde3 f 2 l − 144c2 e2 f 3 l + 20de3 f 3 l − 20ce2 f 4 l − 20c2 de4 m + 4d2 e5 m + 70c3 e3 f m − 18cde4 f m + 64c2 e3 f 2 m − 10de4 f 2 m + 10ce3 f 3 m − 10c3 e4 n + 2cde5 n − 12c2 e4 f n + 2de5 f n − 2ce4 f 2 n − c3 d3 ep + 2cd4 e2 p + c4 d2 f p − 11c2 d3 ef p + 2d4 e2 f p + 9c3 d2 f 2 p − 27cd3 ef 2 p + 25c2 d2 f 3 p − 17d3 ef 3 p + 27cd2 f 4 p + 10d2 f 5 p + 5c4 d2 eq − 12c2 d3 e2 q + 4d4 e3 q − 5c5 df q + 59c3 d2 ef q − 34cd3 e2 f q − 47c4 df 2 q + 171c2 d2 ef 2 q − 34d3 e2 f 2 q − 141c3 df 3 q + 153cd2 ef 3 q − 169c2 df 4 q + 20d2 ef 4 q − 70cdf 5 q + 10c5 der − 42c3 d2 e2 r + 8cd3 e3 r − 10c6 f r + 114c4 def r − 150c2 d2 e2 f r

Appendix 9

199

+ 24d3 e3 f r − 72c5 f 2 r + 246c3 def 2 r − 66cd2 e2 f 2 r − 104c4 f 3 r + 90c2 def 3 r − 6d2 e2 f 3 r + 52c3 f 4 r − 8cdef 4 r + 114c2 f 5 r − 20def 5 r + 20cf 6 r − 10c4 de2 s + 42c2 d2 e3 s − 8d3 e4 s + 10c5 ef s − 104c3 de2 f s + 28cd2 e3 f s + 62c4 ef 2 s − 162c2 de2 f 2 s + 26d2 e3 f 2 s + 42c3 ef 3 s − 32cde2 f 3 s − 94c2 ef 4 s + 20de2 f 4 s − 20cef 5 s + 10c3 de3 u − 2cd2 e4 u − 10c4 e2 f u + 94c2 de3 f u − 18d2 e4 f u − 42c3 e2 f 2 u + 26cde3 f 2 u + 42c2 e2 f 3 u − 10de3 f 3 u + 10ce2 f 4 u − 20c2 de4 v + 4d2 e5 v + 10c3 e3 f v − 6cde4 f v − 8c2 e3 f 2 v + 2de4 f 2 v − 2ce3 f 3 v); G23 = 10(−c3 d3 eg − 2cd4 e2 g + c4 d2 f g − 4c2 d3 ef g − 14d4 e2 f g + 6c3 d2 f 2 g + 21cd3 ef 2 g + 3c2 d2 f 3 g + 20d3 ef 3 g − 10cd2 f 4 g + 5c4 d2 eh + 9c2 d3 e2 h − 5c5 df h + 22c3 d2 ef h + 64cd3 e2 f h − 31c4 df 2 h − 93c2 d2 ef 2 h + 7d3 e2 f 2 h − 21c3 df 3 h − 104cd2 ef 3 h + 47c2 df 4 h − 10d2 ef 4 h + 10cdf 5 h − 10c5 dek − 13c3 d2 e2 k + 4cd3 e3 k + 10c6 f k − 54c4 def k − 105c2 d2 e2 f k + 28d3 e3 f k + 67c5 f 2 k + 128c3 def 2 k − 99cd2 e2 f 2 k + 73c4 f 3 k + 244c2 def 3 k − 47d2 e2 f 3 k − 73c3 f 4 k + 114cdef 4 k − 67c2 f 5 k + 10def 5 k − 10cf 6 k + 40c4 de2 l − 28c2 d2 e3 l − 40c5 ef l + 176c3 de2 f l − 88cd2 e3 f l − 248c4 ef 2 l + 246c2 de2 f 2 l − 12d2 e3 f 2 l − 258c3 ef 3 l + 104cde2 f 3 l − 92c2 ef 4 l + 10de2 f 4 l − 10cef 5 l − 20c3 de3 m + 14cd2 e4 m + 70c4 e2 f m − 63c2 de3 f m + 2d2 e4 f m + 99c3 e2 f 2 m − 44cde3 f 2 m + 42c2 e2 f 3 m − 5de3 f 3 m + 5ce2 f 4 m − 10c4 e3 n + 7c2 de4 n − 17c3 e3 f n + 8cde4 f n − 8c2 e3 f 2 n + de4 f 2 n − ce3 f 3 n − c2 d4 ep + c3 d3 f p − 8cd4 ef p + 8c2 d3 f 2 p − 7d4 ef 2 p + 17cd3 f 3 p + 10d3 f 4 p + 5c3 d3 eq − 2cd4 e2 q − 5c4 d2 f q + 44c2 d3 ef q − 14d4 e2 f q − 42c3 d2 f 2 q + 63cd3 ef 2 q − 99c2 d2 f 3 q + 20d3 ef 3 q − 70cd2 f 4 q − 10c4 d2 er + 12c2 d3 e2 r + 10c5 df r − 104c3 d2 ef r + 88cd3 e2 f r + 92c4 df 2 r − 246c2 d2 ef 2 r + 28d3 e2 f 2 r + 258c3 df 3 r − 176cd2 ef 3 r + 248c2 df 4 r − 40d2 ef 4 r + 40cdf 5 r − 10c5 des + 47c3 d2 e2 s − 28cd3 e3 s + 10c6 f s − 114c4 def s + 99c2 d2 e2 f s − 4d3 e3 f s + 67c5 f 2 s − 244c3 def 2 s + 105cd2 e2 f 2 s + 73c4 f 3 s − 128c2 def 3 s + 13d2 e2 f 3 s − 73c3 f 4 s + 54cdef 4 s − 67c2 f 5 s + 10def 5 s − 10cf 6 s + 10c4 de2 u − 7c2 d2 e3 u − 10c5 ef u + 104c3 de2 f u − 64cd2 e3 f u − 47c4 ef 2 u + 93c2 de2 f 2 u − 9d2 e3 f 2 u + 21c3 ef 3 u − 22cde2 f 3 u + 31c2 ef 4 u − 5de2 f 4 u + 5cef 5 u − 20c3 de3 v + 14cd2 e4 v + 10c4 e2 f v − 21c2 de3 f v

200

Appendix 9 + 2d2 e4 f v − 3c3 e2 f 2 v + 4cde3 f 2 v − 6c2 e2 f 3 v + de3 f 3 v − ce2 f 4 v);

G24 = 5(2c2 d4 eg + 4d5 e2 g − 2c3 d3 f g − 6cd4 ef g − 8c2 d3 f 2 g − 20d4 ef 2 g + 10cd3 f 3 g − 10c3 d3 eh − 18cd4 e2 h + 10c4 d2 f h + 26c2 d3 ef h − 2d4 e2 f h + 42c3 d2 f 2 h + 94cd3 ef 2 h − 42c2 d2 f 3 h + 10d3 ef 3 h − 10cd2 f 4 h + 20c4 d2 ek + 26c2 d3 e2 k − 8d4 e3 k − 20c5 df k − 32c3 d2 ef k + 28cd3 e2 f k − 94c4 df 2 k − 162c2 d2 ef 2 k + 42d3 e2 f 2 k + 42c3 df 3 k − 104cd2 ef 3 k + 62c2 df 4 k − 10d2 ef 4 k + 10cdf 5 k − 20c5 del − 6c3 d2 e2 l + 24cd3 e3 l + 20c6 f l − 8c4 def l − 66c2 d2 e2 f l + 8d3 e3 f l + 114c5 f 2 l + 90c3 def 2 l − 150cd2 e2 f 2 l + 52c4 f 3 l + 246c2 def 3 l − 42d2 e2 f 3 l − 104c3 f 4 l + 114cdef 4 l − 72c2 f 5 l + 10def 5 l − 10cf 6 l + 20c4 de2 m − 34c2 d2 e3 m + 4d3 e4 m − 70c5 ef m + 153c3 de2 f m − 34cd2 e3 f m − 169c4 ef 2 m + 171c2 de2 f 2 m − 12d2 e3 f 2 m − 141c3 ef 3 m + 59cde2 f 3 m − 47c2 ef 4 m + 5de2 f 4 m − 5cef 5 m + 10c5 e2 n − 17c3 de3 n + 2cd2 e4 n + 27c4 e2 f n − 27c2 de3 f n + 2d2 e4 f n + 25c3 e2 f 2 n − 11cde3 f 2 n + 9c2 e2 f 3 n − de3 f 3 n + ce2 f 4 n + 2cd5 ep − 2c2 d4 f p + 2d5 ef p − 12cd4 f 2 p − 10d4 f 3 p − 10c2 d4 eq + 4d5 e2 q + 10c3 d3 f q − 18cd4 ef q + 64c2 d3 f 2 q − 20d4 ef 2 q + 70cd3 f 3 q + 20c3 d3 er − 24cd4 e2 r − 20c4 d2 f r + 68c2 d3 ef r − 8d4 e2 f r − 144c3 d2 f 2 r + 136cd3 ef 2 r − 228c2 d2 f 3 r + 40d3 ef 3 r − 40cd2 f 4 r − 20c4 d2 es + 54c2 d3 e2 s − 8d4 e3 s + 20c5 df s − 128c3 d2 ef s + 60cd3 e2 f s + 174c4 df 2 s − 366c2 d2 ef 2 s + 46d3 e2 f 2 s + 414c3 df 3 s − 272cd2 ef 3 s + 226c2 df 4 s − 30d2 ef 4 s + 30cdf 5 s − 10c5 deu + 17c3 d2 e2 u − 2cd3 e3 u + 10c6 f u − 124c4 def u + 165c2 d2 e2 f u − 18d3 e3 f u + 57c5 f 2 u − 239c3 def 2 u + 99cd2 e2 f 2 u + 26c4 f 3 u − 29c2 def 3 u − d2 e2 f 3 u − 52c3 f 4 u + 37cdef 4 u − 36c2 f 5 u + 5def 5 u − 5cf 6 u + 20c4 de2 v − 34c2 d2 e3 v + 4d3 e4 v − 10c5 ef v + 51c3 de2 f v − 22cd2 e3 f v − 7c4 ef 2 v + 9c2 de2 f 2 v + 9c3 ef 3 v − 7cde2 f 3 v + 7c2 ef 4 v − de2 f 4 v + cef 5 v); G25 = −10c2 d4 f g − 20d5 ef g + 10cd4 f 2 g + 50c3 d3 f h + 90cd4 ef h − 40c2 d3 f 2 h + 10d4 ef 2 h − 10cd3 f 3 h − 100c4 d2 f k − 130c2 d3 ef k + 40d4 e2 f k + 30c3 d2 f 2 k − 100cd3 ef 2 k + 60c2 d2 f 3 k − 10d3 ef 3 k + 10cd2 f 4 k + 100c5 df l + 30c3 d2 ef l − 120cd3 e2 f l + 70c4 df 2 l + 210c2 d2 ef 2 l − 40d3 e2 f 2 l − 90c3 df 3 l + 110cd2 ef 3 l − 70c2 df 4 l + 10d2 ef 4 l − 10cdf 5 l − 50c6 f m + 85c4 def m + 90c2 d2 e2 f m − 20d3 e3 f m

Appendix 9

201

− 135c5 f 2 m − 65c3 def 2 m + 150cd2 e2 f 2 m − 25c4 f 3 m − 255c2 def 3 m + 40d2 e2 f 3 m + 125c3 f 4 m − 115cdef 4 m + 75c2 f 5 m − 10def 5 m + 10cf 6 m + 10c6 en − 37c4 de2 n + 16c2 d2 e3 n + 47c5 ef n − 95c3 de2 f n + 22cd2 e3 f n + 79c4 ef 2 n − 81c2 de2 f 2 n + 6d2 e3 f 2 n + 59c3 ef 3 n − 25cde2 f 3 n + 19c2 ef 4 n − 2de2 f 4 n + 2cef 5 n − 10cd5 f p − 10d5 f 2 p + 50c2 d4 f q − 20d5 ef q + 70cd4 f 2 q − 100c3 d3 f r + 120cd4 ef r − 220c2 d3 f 2 r + 40d4 ef 2 r − 40cd3 f 3 r + 100c4 d2 f s − 270c2 d3 ef s + 40d4 e2 f s + 370c3 d2 f 2 s − 260cd3 ef 2 s + 220c2 d2 f 3 s − 30d3 ef 3 s + 30cd2 f 4 s − 50c5 df u + 285c3 d2 ef u − 150cd3 e2 f u − 335c4 df 2 u + 615c2 d2 ef 2 u − 70d3 e2 f 2 u − 465c3 df 3 u + 275cd2 ef 3 u − 205c2 df 4 u + 25d2 ef 4 u − 25cdf 5 u + 20c5 dev − 74c3 d2 e2 v + 32cd3 e3 v − 10c6 f v + 111c4 def v − 132c2 d2 e2 f v + 12d3 e3 f v − 27c5 f 2 v + 105c3 def 2 v − 36cd2 e2 f 2 v − 5c4 f 3 v − c2 def 3 v + 2d2 e2 f 3 v + 25c3 f 4 v − 17cdef 4 v + 15c2 f 5 v − 2def 5 v + 2cf 6 v; G26 = 10c2 d5 g + 20d6 eg − 10cd5 f g − 50c3 d4 h − 90cd5 eh + 40c2 d4 f h − 10d5 ef h + 10cd4 f 2 h + 100c4 d3 k + 130c2 d4 ek − 40d5 e2 k − 30c3 d3 f k + 100cd4 ef k − 60c2 d3 f 2 k + 10d4 ef 2 k − 10cd3 f 3 k − 100c5 d2 l − 30c3 d3 el + 120cd4 e2 l − 70c4 d2 f l − 210c2 d3 ef l + 40d4 e2 f l + 90c3 d2 f 2 l − 110cd3 ef 2 l + 70c2 d2 f 3 l − 10d3 ef 3 l + 10cd2 f 4 l + 50c6 dm − 85c4 d2 em − 90c2 d3 e2 m + 20d4 e3 m + 135c5 df m + 65c3 d2 ef m − 150cd3 e2 f m + 25c4 df 2 m + 255c2 d2 ef 2 m − 40d3 e2 f 2 m − 125c3 df 3 m + 115cd2 ef 3 m − 75c2 df 4 m + 10d2 ef 4 m − 10cdf 5 m − 10c7 n + 67c5 den − 27c3 d2 e2 n − 22cd3 e3 n − 77c6 f n + 167c4 def n + 81c2 d2 e2 f n − 22d3 e3 f n − 140c5 f 2 n − 59c3 def 2 n + 147cd2 e2 f 2 n − 265c2 def 3 n + 39d2 e2 f 3 n + 140c3 f 4 n − 116cdef 4 n + 77c2 f 5 n − 10def 5 n + 10cf 6 n + 10cd6 p + 10d6 f p − 50c2 d5 q + 20d6 eq − 70cd5 f q + 100c3 d4 r − 120cd5 er + 220c2 d4 f r − 40d5 ef r + 40cd4 f 2 r − 100c4 d3 s + 270c2 d4 es − 40d5 e2 s − 370c3 d3 f s + 260cd4 ef s − 220c2 d3 f 2 s + 30d4 ef 2 s − 30cd3 f 3 s + 50c5 d2 u − 285c3 d3 eu + 150cd4 e2 u + 335c4 d2 f u − 615c2 d3 ef u + 70d4 e2 f u + 465c3 d2 f 2 u − 275cd3 ef 2 u + 205c2 d2 f 3 u − 25d3 ef 3 u + 25cd2 f 4 u − 10c6 dv + 137c4 d2 ev − 186c2 d3 e2 v + 20d4 e3 v − 147c5 df v + 635c3 d2 ef v − 282cd3 e2 f v − 449c4 df 2 v + 741c2 d2 ef 2 v − 76d3 e2 f 2 v − 479c3 df 3 v+

202

Appendix 9 265cd2 ef 3 v − 189c2 df 4 v + 22d2 ef 4 v − 22cdf 5 v; σ20 = 2e3 (de − cf )(−5c2 + 16de − 26cf − 5f 2 )(−2c2 + de − 5cf − 2f 2 ); σ21 = e2 (c − f )(de − cf )(−5c2 + 16de − 26cf − 5f 2 )(−2c2 + de − 5cf − 2f 2 ); σ22 = 2e(de − cf )(−5c2 + 16de − 26cf − 5f 2 )(−2c2 + de − 5cf − 2f 2 )· · (−c2 + de + 2cf − f 2 ); σ23 = (c − f )(−de + cf )(c2 − 6de − 2cf + f 2 )(2c2 − de + 5cf + 2f 2 )· · (5c2 − 16de + 26cf + 5f 2 ); σ24 = 2d(de − cf )(−5c2 + 16de − 26cf − 5f 2 )(−2c2 + de − 5cf − 2f 2 )· · (−c2 + de + 2cf − f 2 ); σ25 = d2 (c − f )(de − cf )(−5c2 + 16de − 26cf − 5f 2 )(−2c2 + de − 5cf − 2f 2 ); σ26 = 2d3 (de − cf )(−5c2 + 16de − 26cf − 5f 2 )(−2c2 + de − 5cf − 2f 2 );

D20 = −60c5 de + 222c3 d2 e2 − 96cd3 e3 + 60c6 f − 744c4 def + 954c2 d2 e2 f − 96d3 e3 f + 522c5 f 2 − 2220c3 def 2 + 954cd2 e2 f 2 + 1362c4 f 3 − 2220c2 def 3 + 222d2 e2 f 3 + 1362c3 f 4 − 744cdef 4 + 522c2 f 5 − 60def 5 + 60cf 6 ; D21 = 60c5 de − 222c3 d2 e2 + 96cd3 e3 − 60c6 f + 744c4 def − 954c2 d2 e2 f + 96d3 e3 f − 522c5 f 2 + 2220c3 def 2 − 954cd2 e2 f 2 − 1362c4 f 3 + 2220c2 def 3 − 222d2 e2 f 3 − 1362c3 f 4 + 744cdef 4 − 522c2 f 5 + 60def 5 − 60cf 6 ; D22 = 5(60c5 de − 222c3 d2 e2 + 96cd3 e3 − 60c6 f + 744c4 def − 954c2 d2 e2 f + 96d3 e3 f − 522c5 f 2 + 2220c3 def 2 − 954cd2 e2 f 2 − 1362c4 f 3 + 2220c2 def 3 − 222d2 e2 f 3 − 1362c3 f 4 + 744cdef 4 − 522c2 f 5 + 60def 5 − 60cf 6 ); D23 = 10(−60c5 de + 222c3 d2 e2 − 96cd3 e3 + 60c6 f − 744c4 def + 954c2 d2 e2 f − 96d3 e3 f + 522c5 f 2 − 2220c3 def 2 + 954cd2 e2 f 2 + 1362c4 f 3 − 2220c2 def 3 + 222d2 e2 f 3 + 1362c3 f 4 − 744cdef 4 + 522c2 f 5 − 60def 5 + 60cf 6 ); D24 = 5(−60c5 de + 222c3 d2 e2 − 96cd3 e3 + 60c6 f − 744c4 def + 954c2 d2 e2 f − 96d3 e3 f + 522c5 f 2 − 2220c3 def 2 + 954cd2 e2 f 2 + 1362c4 f 3 − 2220c2 def 3 + 222d2 e2 f 3 + 1362c3 f 4 − 744cdef 4 + 522c2 f 5 − 60def 5 + 60cf 6 );

Appendix 9 D25 = 60c5 de − 222c3 d2 e2 + 96cd3 e3 − 60c6 f + 744c4 def − 954c2 d2 e2 f + 96d3 e3 f − 522c5 f 2 + 2220c3 def 2 − 954cd2 e2 f 2 − 1362c4 f 3 + 2220c2 def 3 − 222d2 e2 f 3 − 1362c3 f 4 + 744cdef 4 − 522c2 f 5 + 60def 5 − 60cf 6 ; D26 = 60c5 de − 222c3 d2 e2 + 96cd3 e3 − 60c6 f + 744c4 def − 954c2 d2 e2 f + 96d3 e3 f − 522c5 f 2 + 2220c3 def 2 − 954cd2 e2 f 2 − 1362c4 f 3 + 2220c2 def 3 − 222d2 e2 f 3 − 1362c3 f 4 + 744cdef 4 − 522c2 f 5 + 60def 5 − 60cf 6 .

203

Appendix 10

e2 B e2 = C e2 for Matrices that Define a System of Linear Equations A the Quantity G2 in Case of the Differential System s(1, 2, 3) e2 = [A e02 |A e002 |A e000 e0000 e00000 A 2 |A2 |A2 ],

                    0 e A2 =                    

3c 3e 3d 6c + 3f 0 6d 0 0 3g 3l 6h 6g + 6m 3k 12h + 3n 0 6k 0 0 3p 3t 9q 6p + 9u 9r 18q + 9v 3s 18r + 3w 0 6s 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 6e 3c + 6f 3d 0 6l 3g + 12m 6h + 6n 3k 0 6t 3p + 18u 9q + 18v 9r + 6w 3s 0 0 0 0 0 0 0

0 0 0 0 3e 0 3f 0 0 4c 0 4d 3l 0 6m 0 3n 0 0 4g 0 8h 3t 4k 9u 0 9v 0 3w 0 0 4p 0 12q 0 12r 0 4s 0 0 0 0 0 0

 0 0  0 0   0 0   0 0   4e 0   12c + 4f 12e  12d 12c + 12f    0 12d   0 0   4l 0   12g + 8m 12l , 24h + 4n 12g + 24m   12k 24h + 12n    0 12k   0 0   4t 0   12p + 12u 12t  36q + 12v 12p + 36u   36r + 4w 36q + 36v   12s 36r + 12w    0 12s 0 0

205

206                     e00 =  A 2                   

0 0 0 0 0 0 12e 4c + 12f 4d 0 0 12l 4g + 24m 8h + 12n 4k 0 0 12t 4p + 36u 12q + 36v 12r + 12w 4s 

                   000 e A2 =                    

0 0 0 0 0 0 0 0 0 −e2 0 2ce − 2ef 0 −c2 + 2de + 2cf − f 2 4e −2cd + 2df 4f −d2 0 0 0 0 0 0 4l 0 8m 0 4n 0 0 0 0 0 0 0 4t 0 12u 0 12v 0 4w 0

0 0 0 0 0 0 0 0 0 0 20e 30c + 20f 30d 0 0 0 20l 30g + 40m 60h + 20n 30k 0 0

0 0 0 0 0 0 0 0 0 0 0 30e 20c + 30f 20d 0 0 0 30l 20g + 60m 40h + 30n 20k 0

0 0 0 0 0 0 0 0 0 0 0 0 20e 5c + 20f 5d 0 0 0 20l 5g + 40m 10h + 20n 5k

Appendix 10  0 0  0 0   0 0   0 0   0 0   0 0   0 0   0 0   0 0   5c 5e  5d 20c + 5f  ,  0 20d   0 0   0 0   0 0   5g 5l  10h 20g + 10m   5k 40h + 5n    0 20k   0 0   0 0 0 0  0 0 0 0   0 0   0 0   0 0   0 0   0 0   0 0   0 0   0 0   0 0  , 0 0   0 0   5e 0   5f 0   0 6c   0 6d   0 0   0 0   5l 0   10m 0  5n 0

Appendix 10 

e0000 A 2

207

0  0   0   0   0   0   0   0   0   0   0 =  0   0   0   0   6e   30c + 6f   30d   0   0   0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30e 60c + 30f 60d 0 0 0 

e00000 A 2

                   =                   

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6e 6f

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60e 60c + 60f 60d 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60e 30c + 60f 30d 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 e3 3e2 (f − c) 3e[(c + f )2 − de] −(c − f )[(c − f )2 − 6de] 3d[−(c − f )2 + de] 3d2 (f − c) −d3

                    ,                   

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30e 6c + 30f 6d

                    ,                   

208

Appendix 10                       e B2 =                      

a0 a1 a2 a3 b0 b1 b2 b3 b4 G1 c0 c1 c2 c3 c4 c5 d0 d1 d2 d3 d4 d5 d6 G2

                      ,                     

                    e C2 =                    

2eg − cl + f l −cg + f g + 4eh − 2dl − 2cm + 2f m −2ch + 2f h + 2ek − 4dm − cn + f n −ck + f k − 2dn 2ep − ct + f t −cp + f p + 6eq − 2dt − 3cu + 3f u −3cq + 3f q + 6er − 6du − 3cv + 3f v −3cr + 3f r + 2es − 6dv − cw + f w −cs + f s − 2dw 0 0 0 0 0 0 0 0 0 0 0 0 0

                    .                   

Bibliography

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Index

Afraimovich V. S. 207 Alekseev V. G. 207 algebraic bases 37, 39, 40, 41, 42, 44, 62, 63, 210 algebraically independent 3, 37, 62, 63, 144 focus quantities 3, 4, 59, 98, 101, 110, 113, 123, 135, 144, 146, 147, 149, 162, 164 generalized focus-quantities 98, 101, 110, 113, 123, 127, 135, 138, 144, 145 Andrunachievici V. A. 5 Arnold V. I. 207 Arzhantsev I. V. 207 Baltag V. A. 2, 207 Bautin N. N. 207, 210 Bernoulli D. 57 Bothmer H. C. 208 Calin Iu. F. 2, 159, 161, 207 Cayley A. 49, 50, 57, 66, 68, 70, 72 center and focus problem 1, 2, 3, 4, 59, 81, 82, 83, 84, 86, 87, 98, 101, 109, 110, 113, 123, 127, 135, 138, 144, 145, 146, 147, 149, 156, 159, 162, 163, 209 centro-affine 4 comitants 4, 23, 25, 26, 29, 35, 36, 37, 38, 39, 44, 45, 46, 47, 49, 50, 56, 57, 81, 82, 86, 92, 95, 98, 106, 109, 120, 121, 123, 133, 135, 153 group 4, 7, 19, 23, 38, 39, 40, 147

invariants 4, 23, 25, 26, 29, 30, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 57, 82, 84, 86, 153, 210 transformations 11, 12, 26, 27, 29, 46, 83, 155, 163 Ciobanu M. 5, 207 Ciubotaru S. 207 comitant under the rotation group 151 Cozma D. V. 2, 5, 208 Derksen H 208 Dub´e Thomas 208 essential center conditions 3, 163 Euler L. 32, 57 focus pseudo-quantity 97, 98, 101, 109, 110, 123, 135, 138, 140, 141, 143, 144, 145, 146, 175, 187, 197 focus quantities 3, 4, 5, 81, 82, 83, 84, 87, 95, 98, 101, 110, 113, 123, 127, 135, 138, 144, 146, 147, 149, 156, 158, 159, 161, 162, 163, 164 Franklin F. 208 functional basis 151 Gauss K. 57 the generalized center and focus problem 3, 4, 82, 86, 147 generating functions 4, 45, 47, 49, 50, 51, 54, 57, 58, 59, 65, 68, 70, 72, 77, 78, 87, 123, 124, 135, 136, 208, 209 Gher¸stega N. N. 208 213

214 graded algebras 55, 56, 65, 87, 101, 113, 127, 138, 208 Graf V. 208 Gurevich G. B. 208 Haar measure 74 Hilbert D. 57, 62 Hilbert series 4, 45, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 77, 78, 79, 87, 98, 100, 110, 112, 123, 124, 125, 126, 135, 136, 137, 147, 209 Hilbert Theorem 3 hypothesis 3, 127, 138, 146, 147, 162, 163, 209 Ilyashenko Yu. S. 207, 208 invariant under the rotation group 151, 152, 158 Kemper Gregor 208 Kr¨ oker J. 208 Krull dimension 4, 5, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 78, 79, 145, 147, 164 Laurent 75 Lie algebra 4, 7, 19, 21, 22, 29, 30, 35, 38, 39, 41, 61, 91, 105, 131, 140, 147, 209 Lie operator 14, 15, 19, 149, 150, 151, 152, 153, 154, 156, 158, 159, 162 Lunkevich V. A. 208 Lyapunov A. M. 2, 3, 81, 208 Lyapunov system 4, 82, 149, 151, 152, 154, 155, 156, 162, 163 Macari P. M. 210 the main result of this book 4, 164 Markushevich A. I. 209 Molien’s formula 74 Newton I. 57 null focus pseudo-quantity 83, 84 one-parameter group 10, 13, 14, 15, 19, 22, 150

Index Ovsyannikov L. V. 19, 209 Poincar´e H. 1, 2, 3, 59, 81, 147, 207, 209 Poincar´e-Lyapunov’s constants 2 pole 58, 64, 67, 68, 74, 76, 77 Pontryagin L. S. 209 Popa M. N. 208, 209, 210 Pricop V. V. 208, 209 Repeco V. 209 Residue Theorem 75, 76, 77 Riemann R. 57 Rotaru T. 5, 207 rotation group 149, 151, 152, 153, 154, 155, 156, 158, 159, 161, 162 r -parameter group 11, 14 Sadovsky A. P. 156, 158, 210 Saint Luke (Voyno-Yasenetsky) 210 semi-invariant 37, 86, 96, 97, 98, 107, 109, 110, 121, 123, 134, 135, 144, 145, 156, 159, 161, 162 Shilnikov L. P. 207 Sibirsky algebra 4, 5, 57, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 78, 79, 92, 95, 96, 97, 98, 106, 107, 109, 110, 123, 135, 141, 144, 147, 152, 164 Sibirsky invariant variety 82, 83, 86, 92, 94, 95, 106, 121, 133, 145 Sibirsky K. S. 2, 57, 81, 208, 210 singular point 1, 2, 3, 81, 83, 84, 146, 147, 163 center 1, 2, 3, 4, 5, 59, 81, 82, 83, 84, 86, 87, 95, 98, 101, 109, 110, 113, 123, 127, 135, 138, 144, 145, 146, 147, 149, 156, 159, 162, 163, 207, 208, 209 focus 1, 2, 3, 4, 5, 59, 81, 82, 83, 84, 86, 87, 92, 95, 96,

Index 97, 98, 101, 109, 110, 113, 123, 127, 135, 138, 140, 142, 143, 144, 145, 146, 147, 149, 156, 158, 159, 161, 162, 163, 164, 207, 209 node 1 saddle 1 Springer T. A. 210 S ¸ ub˘ a A. S. 2, 5 Sylvester D. 50, 51, 57, 66, 68, 70, 72

215 syzygy 39, 41, 43, 61 Ufnarovskij V. A. 210 unimodular 11 comitants 55, 56, 57, 64, 87, 101, 113, 127, 138, 145 group 12, 145 invariants 55, 64 transformations 11, 12, 55 Vulpe N. I. 2, 5, 210 weight isobarity 84, 86 ˙ adek H. 3, 210 Zol¸

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