An Introduction to Gröbner Bases [1 ed.] 0471974420, 9780471974420

Table of contents :
Cover
Title
Contents
Preface
1 Rings, Fields, and Ideals
1.1 Definition of a Ring
Exercises
1.2 Rings and Fields
1.2.1 Integral Domains and Fields
1.2.2 A Convention
1.2.3 A Finite Ring, Z
1.2.4 Polynomial Rings
1.2.5 Zerodivisors
1.3 Ideals and Rings
1.3.1 Definition of an Ideal
1.3.2 The Ideal Generated by a Set
1.3.3 Principal Ideals and Euclid's Algorithm
1.3.4 Euclidean Rings
1.3.5 Ideals and their Calculus
1.4 Equivalence Relations
1.5 Field of Fractions of an Integral Domain
Exercises
1.6 Unique Factorization Domains
Exercises
1.7 Factor Rings and Homomorphisms
1.8 Prime Ideals and Maximal Ideals
1.9 Vector Spaces
Exercises
2 Monomial Ideals
2.1 Sums and Products of Monomial Ideals
2.2 Intersections of Monomial Ideals
2.3 Quotients of Monomial Ideals
2.4 Prime Ideals
2.5 The Radical of a Monomial Ideal
3 Grobner Bases
3.1 Monomial Orderings
3.1.1 A Classification of Orderings
3.2 Dickson's Lemma and Some Applications
3.2.1 Dickson's Lemma
3.2.2 Applications of Dickson's Lemma
3.3 The Reduction Process
3.4 Definition of Grobner Bases
3.5 Hubert's Basis Theorem and Noetherian Rings
3.6 Grobner Bases and Normal Forms
3.7 Reduced Grobner Bases
3.8 Construction of Grobner Bases
3.9 Free Modules and Syzygies
3.10 Syzygies of Sequences of Monomials
3.11 S-polynomials
3.12 A Criterion for Grobner Basis
3.13 The Buchberger Algorithm
4 Algebraic Sets
4.1 Algebraic Sets and Ideals
4.2 Hilbert's Nullstellensatz
4.3 A Dictionary: Algebraic Sets ** Radical Ideals
5 Primary Decomposition of Ideals
6 Solving Systems of Polynomial Equations
6.1 Systems with Only One Solution
Exercises
6.2 Systems with Finitely Many Solutions
6.2.1 Decomposition of the Ideal
6.2.2 Decomposition of the Ring
6.3 Solving Zero-dimensional Systems
Exercises
6.4 Systems of Higher Dimension
Exercises
7 Applications of Grobner Bases
7.1 Membership Problems
7.1.1 Ideal Membership
7.1.2 Radical Membership
7.1.3 Subalgebra Membership
7.2 Calculation in Factor Rings of Polynomial Rings
7.3 Elimination
7.4 Ideal Operations
7.4.1 Intersection of Ideals
7.4.2 Ideal Quotient
7.5 Supplementary Exercises
8 Homogeneous Algebras
8.1 Homogeneous Ideals and Algebras
Exercises
8.2 Homogenizing and Dehomogenizing
8.2.1 Homogenizing Polynomials
Exercises
8.2.2 Homogenizing Ideals
Exercises
8.2.3 Dehomogenizing Polynomials
8.2.4 Dehomogenizing Ideals
8.2.5 Homogenization versus Dehomogenization
Exercises
8.3 Grobner Bases for Homogeneous Ideals
Exercises
9 Projective Varieties
9.1 Projective Closure of an Algebraic Set
10 The Associated Graded Ring
11 Hubert Series
11.1 Formal Power Series
11.2 Hilbert Series
11.3 Geometric Meaning of Hilbert Series
12 Variations of Grobner Bases
12.1 Grobner Bases for Modules
Exercises
12.2 More General Orderings within More General Rings
12.3 Grobner Bases for Noncommutative Rings
Exercises
12.4 Differential Grobner Bases
12.5 SAGBI Bases
13 Improvements to Buchberger's Algorithm
13.1 Choice of Ordering
13.2 Strategies
13.3 Unnecessary Reductions
13.4 Homogenizing
13.5 Factorizing
13.6 Coefficients
13.7 Using the Hilbert Series
13.7.1 Calculation of Hilbert Series
13.8 Change of Ordering
13.9 Tracing
14 Software
15 Hints to Some Exercises
16 Answers to Exercises
17 Bibliography
17.1 Books
17.2 Articles
Index

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