Foundations of Analysis over Surreal Number Fields 0444702261, 9780444702265, 9780080872520

Table of contents :
100000......Page 1
200001......Page 2
200002......Page 3
200003......Page 4
200004......Page 5
200005......Page 6
200006......Page 7
200007......Page 8
200008......Page 9
200009......Page 10
200010......Page 11
200011......Page 12
200012......Page 13
200013......Page 14
200014......Page 15
200015......Page 16
200016......Page 17
300001......Page 18
300002......Page 19
300003......Page 20
300004......Page 21
300005......Page 22
300006......Page 23
300007......Page 24
300008......Page 25
300009......Page 26
300010......Page 27
300011......Page 28
300012......Page 29
300013......Page 30
300014......Page 31
300015......Page 32
300016......Page 33
300017......Page 34
300018......Page 35
300019......Page 36
300020......Page 37
300021......Page 38
300022......Page 39
300023......Page 40
300024......Page 41
300025......Page 42
300026......Page 43
300027......Page 44
300028......Page 45
300029......Page 46
300030......Page 47
300031......Page 48
300032......Page 49
300033......Page 50
300034......Page 51
300035......Page 52
300036......Page 53
300037......Page 54
300038......Page 55
300039......Page 56
300040......Page 57
300041......Page 58
300042......Page 59
300043......Page 60
300044......Page 61
300045......Page 62
300046......Page 63
300047......Page 64
300048......Page 65
300049......Page 66
300050......Page 67
300051......Page 68
300052......Page 69
300053......Page 70
300054......Page 71
300055......Page 72
300056......Page 73
300057......Page 74
300058......Page 75
300059......Page 76
300060......Page 77
300061......Page 78
300062......Page 79
300063......Page 80
300064......Page 81
300065......Page 82
300066......Page 83
300067......Page 84
300068......Page 85
300069......Page 86
300070......Page 87
300071......Page 88
300072......Page 89
300073......Page 90
300074......Page 91
300075......Page 92
300076......Page 93
300077......Page 94
300078......Page 95
300079......Page 96
300080......Page 97
300081......Page 98
300082......Page 99
300083......Page 100
300084......Page 101
300085......Page 102
300086......Page 103
300087......Page 104
300088......Page 105
300089......Page 106
300090......Page 107
300091......Page 108
300092......Page 109
300093......Page 110
300094......Page 111
300095......Page 112
300096......Page 113
300097......Page 114
300098......Page 115
300099......Page 116
300100......Page 117
300101......Page 118
300102......Page 119
300103......Page 120
300104......Page 121
300105......Page 122
300106......Page 123
300107......Page 124
300108......Page 125
300109......Page 126
300110......Page 127
300111......Page 128
300112......Page 129
300113......Page 130
300114......Page 131
300115......Page 132
300116......Page 133
300117......Page 134
300118......Page 135
300119......Page 136
300120......Page 137
300121......Page 138
300122......Page 139
300123......Page 140
300124......Page 141
300125......Page 142
300126......Page 143
300127......Page 144
300128......Page 145
300129......Page 146
300130......Page 147
300131......Page 148
300132......Page 149
300133......Page 150
300134......Page 151
300135......Page 152
300136......Page 153
300137......Page 154
300138......Page 155
300139......Page 156
300140......Page 157
300141......Page 158
300142......Page 159
300143......Page 160
300144......Page 161
300145......Page 162
300146......Page 163
300147......Page 164
300148......Page 165
300149......Page 166
300150......Page 167
300151......Page 168
300152......Page 169
300153......Page 170
300154......Page 171
300155......Page 172
300156......Page 173
300157......Page 174
300158......Page 175
300159......Page 176
300160......Page 177
300161......Page 178
300162......Page 179
300163......Page 180
300164......Page 181
300165......Page 182
300166......Page 183
300167......Page 184
300168......Page 185
300169......Page 186
300170......Page 187
300171......Page 188
300172......Page 189
300173......Page 190
300174......Page 191
300175......Page 192
300176......Page 193
300177......Page 194
300178......Page 195
300179......Page 196
300180......Page 197
300181......Page 198
300182......Page 199
300183......Page 200
300184......Page 201
300185......Page 202
300186......Page 203
300187......Page 204
300188......Page 205
300189......Page 206
300190......Page 207
300191......Page 208
300192......Page 209
300193......Page 210
300194......Page 211
300195......Page 212
300196......Page 213
300197......Page 214
300198......Page 215
300199......Page 216
300200......Page 217
300201......Page 218
300202......Page 219
300203......Page 220
300204......Page 221
300205......Page 222
300206......Page 223
300207......Page 224
300208......Page 225
300209......Page 226
300210......Page 227
300211......Page 228
300212......Page 229
300213......Page 230
300214......Page 231
300215......Page 232
300216......Page 233
300217......Page 234
300218......Page 235
300219......Page 236
300220......Page 237
300221......Page 238
300222......Page 239
300223......Page 240
300224......Page 241
300225......Page 242
300226......Page 243
300227......Page 244
300228......Page 245
300229......Page 246
300230......Page 247
300231......Page 248
300232......Page 249
300233......Page 250
300234......Page 251
300235......Page 252
300236......Page 253
300237......Page 254
300238......Page 255
300239......Page 256
300240......Page 257
300241......Page 258
300242......Page 259
300243......Page 260
300244......Page 261
300245......Page 262
300246......Page 263
300247......Page 264
300248......Page 265
300249......Page 266
300250......Page 267
300251......Page 268
300252......Page 269
300253......Page 270
300254......Page 271
300255......Page 272
300256......Page 273
300257......Page 274
300258......Page 275
300259......Page 276
300260......Page 277
300261......Page 278
300262......Page 279
300263......Page 280
300264......Page 281
300265......Page 282
300266......Page 283
300267......Page 284
300268......Page 285
300269......Page 286
300270......Page 287
300271......Page 288
300272......Page 289
300273......Page 290
300274......Page 291
300275......Page 292
300276......Page 293
300277......Page 294
300278......Page 295
300279......Page 296
300280......Page 297
300281......Page 298
300282......Page 299
300283......Page 300
300284......Page 301
300285......Page 302
300286......Page 303
300287......Page 304
300288......Page 305
300289......Page 306
300290......Page 307
300291......Page 308
300292......Page 309
300293......Page 310
300294......Page 311
300295......Page 312
300296......Page 313
300297......Page 314
300298......Page 315
300299......Page 316
300300......Page 317
300301......Page 318
300302......Page 319
300303......Page 320
300304......Page 321
300305......Page 322
300306......Page 323
300307......Page 324
300308......Page 325
300309......Page 326
300310......Page 327
300311......Page 328
300312......Page 329
300313......Page 330
300314......Page 331
300315......Page 332
300316......Page 333
300317......Page 334
300318......Page 335
300319......Page 336
300320......Page 337
300321......Page 338
300322......Page 339
300323......Page 340
300324......Page 341
300325......Page 342
300326......Page 343
300327......Page 344
300328......Page 345
300329......Page 346
300330......Page 347
300331......Page 348
300332......Page 349
300333......Page 350
300334......Page 351
300335......Page 352
300336......Page 353
300337......Page 354
300338......Page 355
300339......Page 356
300340......Page 357
300341......Page 358
300342......Page 359
300343......Page 360
300344......Page 361
300345......Page 362
300346......Page 363
300347......Page 364
300348......Page 365
300349......Page 366
300350......Page 367
300351......Page 368
300352......Page 369
300353......Page 370
300354......Page 371
300355......Page 372
300356......Page 373
300357......Page 374
300358......Page 375
300359......Page 376
300360......Page 377
300361......Page 378
300362......Page 379
300363......Page 380
300364......Page 381
300365......Page 382
300366......Page 383
300367......Page 384
300368......Page 385
300369......Page 386
300370......Page 387
300371......Page 388
300372......Page 389
300373......Page 390
300374......Page 391

Citation preview

FOUNDATIONS OF ANALYSIS OVER SURREAL NUMBER FIELDS

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (117)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro and University of Rochester

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

141

FOUNDATIONS OF ANALYSIS OVER SURREAL NUMBER FIELDS Norman L. ALLING University of Rochester Rochester, NY 14627, U S.A.

1987

NORTH-HOLLAND -AMSTERDAM

0

NEW YORK

0

OXFORD 0 TOKYO

Elsevier Science Publishers B.V., 1987 AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70226 1

Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributorsforthe U.S.A. and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017 U.S.A

PRINTED IN THE NETHERLANDS

For H. L. Alling

This Page Intentionally Left Blank

vi i

PREFACE

I t i s well-known t h a t t h e f i e l d R of a l l r e a l numbers i s a real-

c l o s e d f i e l d and t h a t , up t o iscmorphism, i t i s t h e o n l y Dedekind-complete ordered field.

A r t i n and S c h r e i e r g e n e r a l i z e d t h e a l g e b r a i c p r o p e r t i e s of

the r e a l s t o form t h e r i c h , i n t e r e s t i n g t h e o r y o r r e a l - c l o s e d f i e l d s .

Among o t h e r t h i n g s , t h e y showed t h a t a n y o r d e r e d f i e l d has an a l g e b r a i c extension t h a t i s r e a l - c l o s e d , isomorphism. known.

a n d w h i c h i s u n i q u e l y d e t e r m i n e d up t o

Many i n t e r e s t i n g non-Archimedean, r e a l - c l o s e d f i e l d s F a r e

Under t h e i n t e r v a l t o p o l o g y , a n y o r d e r e d f i e l d i s a t o p o l o g i c a l

field.

Under t h a t t o p o l o g y , F i s n o t Dedekind-complete, i s not l o c a l l y

c o n n e c t e d , and i s not l o c a l l y compact. Using t h e T a r s k i Theorem, we know t h a t every f i r s t o r d e r theorem t h a t is t r u e f o r R is a l s o t r u e f o r any other r e a l - c l o s e d f i e l d , and c o n v e r s e l y .

However, R has many h i g h e r o r d e r p r o p e r t i e s which a r e q u i t e d i f f e r e n t from t h o s e of F.

For example, R i s D e d e k i n d - c o m p l e t e ; a s u b s e t of R i s con-

n e c t e d i f and o n l y i f i t i s a n i n t e r v a l i n R ; and c l o s e d bounded i n t e r v a l s i n R a r e compact.

None of these p r o p e r t i e s a r e t r u e f o r F.

Over t h e l a s t q u a r t e r c e n t u r y , a number of examples of f i e l d s F t h a t are

q

5

- s e t s f o r 6 > 0 have been found.

d e g r e e of d e n s i t y . )

( T h e s e f i e l d s have a v e r y h i g h

However, t h e r e seemed no compelling r e a s o n t o choose

o n e of t h e s e f i e l d s o v e r any o t h e r .

The o n l y n a t u r a l r e g u l a r i z i n g

hypotheses f o r s u c h a f i e l d seemed t o be t h a t ( i ) i t i s o r d e r - i s a n o r p h i c t o H a u s d o r f f ' s normal

q

5

- t y p e , o r ( i i ) t h a t i t i s of power w

5'

While ( i )

seemed r e a s o n a b l e , i t was not c l e a r f o r sane time how s u c h examples could be c o n s t r u c t e d without assuming ( i i ) . Assumption ( i i ) i s e q u i v a l e n t t o a

Norman L. A l l i n g

viii

l o c a l v e r s i o n of t h e g e n e r a l i z e d continuum h y p o t h e s i s ( = G C H ) . appearance of

After t h e

t h e work o f P a u l J . Cohen o n t h e C o n t i n u u m H y p o t h e s i s

(c.19631, t h e GCH seemed, a t l e a s t t o t h e a u t h o r , v e r y f a r f r o m b e i n g a n a t u r a l assumption. Conway p u b l i s h e d

I n 1976 J . H .

0" Numbers

a n d Games, i n w h i c h h e

d e f i n e d a p r o p e r c l a s s No of " n u m b e r s " . w h i c h , t o g e t h e r w i t h i t s r i n g o p e r a t i o n s , was d e f i n e d i n d u c t i v e l y i n o n l y a few i n c i s i v e l i n e s .

He

s u b s e q u e n t l y s k e t c h e d p r o o f s t h a t showed t h a t No is a r e a l - c l o s e d f i e l d . What is much more i m p o r t a n t , i n t h e a u t h o r ' s o p i n i o n , i s t h a t C o n w a y T s f i e l d No h a s some v e r y s t r o n g a d d i t i o n a l p r o p e r t i e s which grow o u t of its construction.

Conway showed t h a t No i s a c o m p l e t e b i n a r y t r e e o f h e i g h t

O n , (On b e i n g t h e c l a s s o f a l l o r d i n a l n u m b e r s ) . numbersT1, a p p l i e d t o No, w a s c o i n e d by D.E.

( T h e term ' ' s u r r e a l

Knuth.)

F o l l o w i n g Conway, we have c a l l e d t h e h e i g h t of an element i n No, i n t h e t r e e s t r u c t u r e on No, i t s b i r t h d a y , a n d t h e h e i g h t s t r u c t u r e o n No i t s b i r t h - o-rder structure. -

One way of s e e i n g j u s t how r i g i d No is, u n d e r i t s

b i r t h - o r d e r s t r u c t u r e , is t h e f o l l o w i n g : No c l e a r l y h a s a g r e a t many f i e l d a u t m o r p h i s m s ; however i t has o n l y o n e b i r t h - o r d e r p r e s e r v i n g a u t a n o r p h i s m . Conway a l s o s u c c e e d e d i n p r o v i n g t h a t No h a s a c a n o n i c a l power s e r i e s structure. Given any o r d i n a l number 5

>

0 , f o r which w

5

is regular, one can

d e f i n e a s u b f i e l d E N 0 o f No, w h i c h h a s a g r e a t many of No's p r o p e r t i e s . For example, gNo is a real-closed f i e l d which is a c o m p l e t e b i n a r y t r e e of height w

5'

F u r t h e r , gNo c a n b e d e s c r i b e d v e r y e a s i l y i n terms of i t s

n a t u r a l formal power series s t r u c t u r e . I t has been known s i n c e a t l e a s t 1960 t h a t any o r d e r e d f i e l d of power

bounded above by w , , c a n be embedded i n a n y r e a l - c l o s e d f i e l d t h a t i s a n 5

n 5- s e t ;

t h u s a l l s u c h f i e l d s may be embedded i n €,No.

With t h i s knowledge in hand t h e a u t h o r d e c i d e d t o t r y t o l e a r n how t o d o a n a l y s i s o v e r 6 1 0 . The p r e s e n t volume i s a r e p o r t on t h e p r o g r e s s , t o d a t e , of t h i s p r o j e c t . More r e s u l t s a r e under s t u d y .

Preface

ix

The f i r s t q u e s t i o n c o n s i d e r e d w a s t h e f o l l o w i n g :

can one modify t h e

i n t e r v a l t o p o l o g y o n CNo i n s u c h a way t h a t t h e r e s u l t i n g "topologyTf has more a t t r a c t i v e p r o p e r t i e s .

The r e s u l t i n g s t r u c t u r e , c a l l e d t h e

t o p o l o g y , i s c o n s i d e r e d a t l e n g t h i n Chapter 2 and 3.

5-

There we f i n d , f o r

example, t h a t t h e c-connected s u b s e t s of CNo a r e e x a c t l y t h e i n t e r v a l s of CNo ( 2 . 2 0 ) .

Conway's book g i v e s an i n s p i r e d s k e t c h of t h e n e c e s s a r y p r o o f s .

On

page 1 7 , h e writes of several of h i s p r o o f s as f o l l o w s : IfProofs l i k e these we c a l l 1 - l i n e p r o o f s e v e n when as h e r e t h e t q l i n e l t i s t o o l o n g f o r o u r We s h a l l meet s t i l l l o n g e r 1 - l i n e p r o o f s l a t e r o n , but t h e y do n o t

pages.

g e t h a r d e r - one s i m p l y t r a n s f o r m s t h e l e f t - h a n d s i d e t h r o u g h t h e d e f i n i t i o n s a n d i n d u c t i v e h y p o t h e s e s u n t i l o n e g e t s t h e r i g h t hand s i d e " .

In

Chapter 4, p a r t of Chapter 5, and a l i t t l e of Chapter 6. we h a v e t r i e d t o c o m p l e t e a l l of Conway's s u g g e s t e d I ' l - l i n e p r o o f s r f , a d d i n g a few new i d e a s h e r e and there.

S i n c e sane v a l u a t i o n t h e o r y seemed t o b e of u s e we h a v e

i n v o k e d q u i t e a l o t of i t .

I n p a r t i c u l a r , t h e t h e o r y o f pseudo-convergent

sequences has been developed and a p p l i e d t o EN0 i n Chapter 6.

We have a l s o

s u p p l i e d a primer of v a l u a t i o n t h e o r y i n Chapter 6. Neumann c o n s i d e r e d formal power s e r i e s , a t a v e r y h i g h

I n 1949 B.H.

l e v e l of g e n e r a l i t y .

Let K be a f i e l d a n d l e t C b e an o r d e r e d Abelian

Let F be t h e f u l l f i e l d K((G)) of formal power series w i t h c o e f f i -

group.

c i e n t s i n K and 71exponents1fi n G.

Let 0 be t h e v a l u a t i o n r i n g of W and l e t M be i t s maxi-

v a l u e group i s C . mal i d e a l .

Let a o ,

one can show t h a t (7.22).

Chapter 7.

F has on i t a n a t u r a l v a l u a t i o n W , whose

... , a n , ... b e

i n K.

Using o n e of Neumann's r e s u l t s ,

&Ioanxn is a w e l l - d e f i n e d

element i n F, f o r a l l XEM,

T h i s we c a l l "Neumann's Theorem", and we g i v e a proof of i t i n Neumann's Theorem c a n e a s i l y b e g e n e r a l i z e d t o c o v e r f o r m a l

power s e r i e s i n s e v e r a l v a r i a b l e s over K ( 7 . 4 1 ) . I t i s not a t a l l d i f f i c u l t t o see t h a t a f o r m a l power s e r i e s f i e l d

e x t e n s i o n of a f o r m a l power s e r i e s f i e l d o v e r K , i s a formal power s e r i e s f i e l d over K ( 7 . 8 0 ) .

What i s p e r h a p s s u r p r i s i n g , a n d i s c e r t a i n l y more

i n t e r e s t i n g , i s t h a t CNo c a n be w r i t t e n as a f o r m a l power series f i e l d

Norman L . A l l i n g

X

e x t e n s i o n of a formal power series f i e l d o v e r R , i n a g r e a t many i n t e r e s t i n g ways (7.81).

The Main Theorem (7.82) i s an a p p l i c a t i o n of t h e s e i d e a s

combined w i t h t h e g e n e r a l i z a t i o n of Neumann's Theorem d e s c r i b e d a b o v e . S t a t e d v e r y r o u g h l y , The Main Theorem asserts t h a t , g i v e n any formal power

...

series A(X,,

,

X n ) i n a f i n i t e number of v a r i a b l e s X 1 ,

...

,

X

n

with

c o e f f i c i e n t s i n LNo, there e x i s t s a non-zero prime i d e a l P i n t h e v a l u a t i o n r i n g 0 of t h e l l f i n i t e l l elements of CNo s u c h t h a t f o r each element ( x , ,

, x n ) i n Pn , A(x~,

A(xl,

... , X n )

... , x n )

i s a w e l l - d e f i n e d element i n CNo.

i s hyper-convergent over P

.. .

We say t h a t

n

I t i s n o t d i f f i c u l t t o show t h a t s u c h theorems as t h e i m p l i c i t func-

t i o n theorem g e n e r a l i z e o v e r f o r m a l power s e r i e s f i e l d s ( 7 . 7 0 - 7 . 7 4 ) . T h e s e r e s u l t s take o n added i n t e r e s t h e r e because of t h e Main Theorem; f o r when t h e Main Theorem a p p l i e s , t h e r e s u l t i n g formal power s e r i e s h a v e nonz e r o r e g i o n s of hyper-convergence. C l e a r l y one can d e f i n e a 5-continuous f u n c t i o n a s b e i n g a n a l y t i c i f l o c a l l y i t s v a l u e s a r e g i v e n by a hyper-convergent formal power s e r i e s . Such d e f i n i t i o n s are made and i n v e s t i g a t e d i n Chapter 8, which s e r v e s a s a primer on t h a t s u b j e c t . Throughout t h e m a n u s c r i p t , g r e a t e f f o r t s have been made t o m a k e t h i s volume f a i r l y s e l f c o n t a i n e d . a r e cited.

Much e x p o s i t i o n i s g i v e n .

Many r e f e r e n c e s

While e x p e r t s may want t o t u r n q u i c k l y t o new r e s u l t s , s t u d e n t s

s h o u l d be a b l e t o f i n d t h e e x p l a n a t i o n of many elementary p o i n t s of i n t e r -

est herein.

On t h e o t h e r h a n d , many new r e s u l t s a r e g i v e n , a n d much

m a t h e m a t i c s i s b r o u g h t t o b e a r on t h e problems a t hand.

As a f u r t h e r a i d

t o t h e r e a d e r , t h e T a b l e of C o n t e n t s is q u i t e d e s c r i p t i v e , and t h e Index is extensive.

N.L.A.

R o c h e s t e r , NY December 1 1 , 1986

xi

TABLE OF CONTENTS

Page

Section PREFACE

vii

TABLE OF CONTENTS

xi

CHAPTER 0 : INTRODUCTION 0.00

The real numbers

1

0.01

q -fields

2

0.02

The 5 - t o p o l o g y o n a n 0 -set

0.03

Conway's f i e l d No of s u r r e a l numbers

3

0.04

V a l u a t i o n t h e o r y a n d s u r r e a l number f i e l d s

5

0.05

Neumann's theorem and hyper-convergence

5

0.06

The main theorem

6

0.07

A p p l i c a t i o n s of t h e main theorem

7

0.10

E x p o s i t i o n v e r s u s research

7

0.11

References and indexing

9

0.20

P r e r e q u i si t e s

9

0.30

Acknowledgements

5

5

3

10

CHAPTER 1 : PRELIMINARIES 1 .OO

Class t h e o r y a n d s e t t h e o r y

13

1.01

O r d e r e d s e t s and o r d e r t y p e s

16

1.02

W e l l - o r d e r e d s e t s : C a n t o r ' s and von Neumann's o r d i n a l numbers

17

xi i

Norman L . Alling

1.03

Equipotent s e t s , choice, and cardinal numbers

20

1.10

The i n t e r v a l topology

23

r e l a t i ve topology

24

1 .ll The

1.20

C u t s and gaps

25

1.30

Cofinal and c o i n i t i a l sets, c h a r a c t e r s and s a t u r a t i o n

28

1.40

rl

-classes 5

31

1.50

Canpact ordered spaces

33

1.60

Ordered Abelian groups

33

1.61

Hahn valuations on ordered groups

40

1.62

Pseudo-convergent sequences i n Abelian groups w i t h valuation

47

1.63

Skeletons, Hahn groups, and extensions of ordered groups

50

1.64

Hahn's embedding theorem

53

1.65

Ordered d i r e c t sums i n 5H

61

1.66

Canplete and incomplete ordered groups

62

1.70

Ordered r i n g s and f i e l d s

63

1 .71

The Artin-Schreier theory of real-closed f i e l d s

66

1.72

Polynomials i n one v a r i a b l e over real-closed f i e l d s

75

1.73

Rational functions i n one v a r i a b l e over real-closed f i e l d s

78

1.74

Rolle's theorem and a p p l i c a t i o n s

82

1.75

Embedding an ordered f i e l d i n a real-closed rl - f i e l d

5

a4

CHAPTER 2 : THE 5-TOPOLOGY 2.00

The interval topology o n an rl - c l a s s

85

2.01

The 5-topology

85

2.02

A comparison of 5-topologies and w -additive spaces

90

5

2.10

5 The 5-topology on ordered sets and c l a s s e s

2.1 1

€,-closed

92

subclasses of X

94

2.12

The r e l a t i v e 5-topology

94

2.13

On t h e possible non-existence of 5-closures and 5 - i n t e r i o r s

96

2.20

The main theorem on 5-connected subspaces of rl - c l a s s e s

97

2.21

That open subclasses of

2.30

The main theorem on E-compact subspaces of rl - c l a s s e s

101

2.31

5-compact subspaces t h a t a r e not E-closed

103

5

E

-classes a r e E-locally connected

E

101

T a b l e of c o n t e n t s

xiii

2.40

c-continuous maps of o r d e r e d c l a s s e s

104

2.41

An a d d i t i o n a l theorem on c-continuous maps

106

CHAPTER 3: THE c-TOPOLOGY ON AFFINE n-SPACE

3.00

The s t r o n g topology and s e m i - a l g e b r a i c s e t s

109

3.10 The a f f i n e l i n e

111

3.20

The c-topology on R n

112

3.21

c-continuous maps between a f f i n e s p a c e s

3.30

c-connected subspaces of CR

3.40

R as a t o p o l o g i c a l f i e l d i n t h e c-topology

3.41

R

3.42

The f i e l d C

3.43

Other examples of c-continuous maps

n

112

n

113 114

as a t o p o l o g i c a l v e c t o r s p a c e over R , i n t h e c-topology =

115 115

R ( i ) , as a topological f i e l d

116

CHAPTER 4: INTRODUCTION TO THE SURREAL FIELD No 4.00

S u r r e a l numbers

4.01

Conway's c o n s t r u c t i o n

117 119

4.02

The Cuesta D u t a r i c o n s t r u c t i o n of No

121

4.03

An a b s t r a c t c h a r a c t e r i z a t i o n of a f u l l class of surreal numbers

127

4.04

S u b t r a c t i o n i n No

4.05

Addition i n No

4.06

M u l t i p l i c a t i o n i n No

131 133 138

4.07

Order and m u l t i p l i c a t i o n i n No

141

4.08

The a s s o c i a t i v e law f o r m u l t i p l i c a t i o n i n No

149

4.09

On numbers g i v e n by r e f i n e m e n t s of ( t i m e l y ) Conway c u t s

152

4.10

P r o p e r t i e s of d i v i s i o n i n No

154

4.20

D i s t i n g u i s h e d s u b c l a s s e s of No

160

4.21

Elements of No having f i n i t e b i r t h d a y

161

4.30

165

MU

x

+

4.40

The map XCNO+ w ENO

4.41

F i n i t e l i n e a r combinations of w -x(l)

168

,

...

1

w

over R

171

xiv

Norman L . A l l i n g

4.50

The sign-expansion

175

4.51

The s t r u c t u r e of Z and t h e sign-expansion

178

4.52

The n e a r e s t common p r e d e c e s s o r of a s u b c l a s s of Z

180

4.53

The t r e e s t r u c t u r e of a f u l l c l a s s of s u r r e a l numbers

182

4.54

The predecessor c u t r e p r e s e n t a t i o n of a s u r r e a l number

183

4.60

A l t e r n a t i v e axioms f o r a f u l l class of s u r r e a l numbers

184

4.61

Conway c u t s , o r d e r e d by e x t e n s i o n , and Cuesta D u t a r i c u t s

189

CHAPTER 5: THE SURREAL FIELDS € N O , AND RELATED TOPICS

5.00

The d e f i n i t i o n of €,No

5.10

€,NO and H a u s d o r f f ' s normal

5.11

The c a r d i n a l number of CNo

5.20

The map XESNO + w EENO

5.30

The s t r u c t u r e of 0 w

x

191 rl

5

-type

192

+

, for

192

193

a l i m i t ordinal

A

195

A

5.40

Rank, u n i v e r s e s , g a l a x i e s , and Conway's c o n s t r u c t i o n

196

5.41

Another d e s c r i p t i o n of CNo

199

5.50

The Dedekind-completion of 0

5.51

The s t r u c t u r e of D

A'

f o r a non-zero l i m i t o r d i n a l A

200 202

A

CHAPTER 6: THE VALUATION THEORY OF ORDERED FIELDS, APPLIED TO NO AND €,NO

Introduction

207

6.01

Examples of f i e l d s w i t h v a l u a t i o n

209

6.10

The v a l u a t i o n t h e o r y of No and SNo

21 1

6.20

Formal power s e r i e s f i e l d s

21 3

6.21

A s k e t c h of Hahn's proof

21 5

6.22

EK(

21 7

6.00

(G)1

and gK((G))

6.23

Algebraic p r o p e r t i e s of K((G))

6.30

Maximal f i e l d s w i t h v a l u a t i o n

21 9

6.40

Pseudo-convergent sequences

221

6.41

Pseudo-convergent sequences i n CNo

223

6.42

Pseudo-convergent sequences i n No

227

21 7

Table of c o n t e n t s

xv

6.43

Normal forms and w-power s e r i e s i n No

6.44

Pseudo-convergent sequences i n K( (C)) and E K ( (C))

232

6.50

Conway's normal form

235

6.51

The i d e n t i t y theorem f o r normal forms i n No

239

6.52

The v e c t o r s p a c e s t r u c t u r e of normal forms

240

6.53

Normal forms i n CNo

242

6.54

M u l t i p l i c a t i o n of normal forms i n No

245

6.55

That t;No i s R-iscmorphic t o a f i e l d of formal power s e r i e s

246

6.56

No a s t h e union of a f a m i l y of formal power series f i e l d s

247

6.57

The c a n o n i c a l n a t u r e of the power s e r i e s s t r u c t u r e on No

248

6.60

That No i s a u n i v e r s a l l y embedding o r d e r e d f i e l d

248

6.70

The i d e a l t h e o r y of v a l u a t i o n r i n g s

250

6.80

B i b l i o g r a p h i c n o t e s on c h a p t e r 6

252

227

CHAPTER 7 : POWER SERIES: FORMAL A N D HYPER-CONVERGENT

7.00

Introduction

255

7.10

Surcomplex number f i e l d s

255

7.11

Cx and formal power series

258

7.20

Neumann' s 1emma

260

7.21

A proof of Neumann's lemma

261

7.22

Neumann's theorem, Neumann s e r i e s , and hyper-convergence

266

7.30

A p p l i c a t i o n s of Neumann's theorem

268

7.31

The a l g e b r a of Neumann s e r i e s

27 1

a formal power s e r i e s f i e l d

7.32

The form of a n i n v e r s e i n

7.33

The binomial series

272

7.34

Powers and v a l u e s of Neumann s e r i e s

275

7.35

C a n p o s i t i o n of Neumann series

27 7

7.36

The e x p o n e n t i a l s e r i e s and t h e l o g a r i t h m i c series

278

7.40

Formal power s e r i e s r i n g s i n a f i n i t e number of v a r i a b l e s

280

7.41

Neumann series i n a f i n i t e number of v a r i a b l e s

28 1

7.50

Trigonometric f u n c t i o n s

28 4

7.51

Elementary f u n c t i o n s over r e a l and complex c o n s t a n t f i e l d s

20 5

7.60

D e r i v a t i v e s of formal power s e r i e s

28 8

7.61

I n f i n i t e s i m a l e x t e n s i o n s of a n a l y t i c f u n c t i o n s , I

289

7.62

The v a l u a t i o n topology

290

272

Norman L . Alling

xvi 7.63

The interval topology and t h e v a l u a t i o n topology

7.64

The modified valuation topology and t h e c-topology on

7.65

I n f i n i t e s i m a l extensions of a n a l y t i c f u n c t i o n s , I1

295

7.70

The formal i m p l i c i t f u n c t i o n theorem i n two v a r i a b l e s

29 6

7.71

The formal i m p l i c i t f u n c t i o n theorem i n n v a r i a b l e s

29 8

7.72

The formal i m p l i c i t mapping l e m m a

301

7.73

The formal i m p l i c i t mapping theorem and t h e Jacobian

303

7.74

The formal inverse mapping theorem

304

7.75

Related theorems on Neumann s e r i e s

306

292 TI

E

-fields

292

7.80

Formal power s e r i e s f i e l d s over formal power s e r i e s f i e l d s

309

7.81

Decomposition of c e r t a i n formal power s e r i e s f i e l d s

31 4

7.82

The main theorem

31 4

7.83

Independence of represent a t i on

31 8

7.84

Prime d i s k s of hyper-convergence of formal power s e r i e s

32 0

7.90

An i n t e r e s t i n g example

32 1

7.91

Fran Maclaurin s e r i e s t o Taylor s e r i e s

322

7.92

Fran Maclaurin s e r i e s t o Taylor s e r i e s over L , I

323

7.93

From Maclaurin s e r i e s t o Taylor s e r i e s over L , I1

327

CHAPTER 8: A PRIMER ON ANALYTIC FUNCTIONS OF A SURREAL VARIABLE

8.00

Introduction

333

8.01

Local p r o p e r t i e s of power s e r i e s i n one v a r i a b l e , I

336

8.02

Local p r o p e r t i e s of power s e r i e s i n one v a r i a b l e , I1

341

8.03

Local p r o p e r t i e s of power s e r i e s i n one v a r i a b l e , I11

3 42

8.04

Local properties of power s e r i e s i n one v a r i a b l e , I V

345

8.05

Local theory of a n a l y t i c functions of one s u r r e a l v a r i a b l e

347

8.10

Local p r o p e r t i e s of power s e r i e s i n s e v e r a l v a r i a b l e s

349

BIBLIOGRAPHY

353

INDEX

359

1

CHAPTER 0

INTRODUCTION

0.00

THE REAL NUMBERS

The f i e l d R of a l l real numbers i s c e n t r a l t o a g r e a t deal of mathe-

matics; s o much s o t h a t i t i s h a r d t o t h i n k of many t o p i c s i n m a t h e m a t i c s w i t h o u t , i n o n e way or an o t h e r , t h i n k i n g about t h e r e a l s .

The f o l l o w i n g

i s well-known: (0)

Up t o isanorphism, R is t h e o n l y Dedekind-complete o r d e r e d f i e l d . One of t h e most s u c c e s s f u l g e n e r a l i z a t i o n s of t h e r e a l s was made by

A r t i n a n d S c h r e i e r i n 1927 [ l o ] ,

i n w h i c h t h e y d e v e l o p e d t h e t h e o r y of

f o r m a l l y real f i e l d s , and of r e a l - c l o s e d f i e l d s .

Thus t h e y g e n e r a l i z e d t h e

a l g e b r a i c t h e o r y of t h e real f i e l d . Given an o r d e r e d ( - l i n e a r l y o r d e r e d ) ( = t o t a l l y o r d e r e d ) g r o u p C , t h e n t h e f o l l o w i n g i s well-known, and w i l l be shown i n due course: (1)

If G is Dedekind-complete t h e n i t is Archimedean, and hence Abelian.

Let K be a n o r d e r e d f i e l d t h a t i s n o t i s a n o r p h i c t o R; t h e n by (01, i t is not Dedekind-complete.




0 , and l e t S

-

{acOn:

Assume t h a t a class X has t h e f o l l o w i n g p r o p e r t i e s : ( i ) OcX; ( i i )

i f acX and i f a

+

1




n x ) ; t h e n L ( x ) and R ( x ) a r e non-

Let A E L ( x ) a n d l e t p e R ( x ) .

m,EZ and n o , n,eN such t h a t

Then

Norman L. A l l i n g

44 ( i ) m,a S n,x,

(9)

Thus, mOnla 5 nonlx

< m,/n,




0 i n G such t h a t

The e q u i v a l e n c e class of o r d e r e d g r o u p s , under o r d e r - p r e s e r v i n g

isomorphisms t h a t p r e s e r v e

0.

I f A = {O) t h e n l e t h ( 0 ) =

Assume now t h a t A f (01.

L e t aEA

S i n c e A i s Archimedean v ( a ) = A , and t h e l a r g e s t proper convex

s u b g r o u p of A is ( 0 ) .

By Theorem 1 , ha i s a homomorphism of A i n t o (R,+)

which p r e s e r v e s S , and has k e r n e l {O).

Thus ha i s a monomorphism of A i n t o

(R,+) which p r e s e r v e s ta f o r

X I ) ,

Thus, i t s u f f i c e s t o prove t h a t x

Let C XI.

t h e n C i s a c u t i n T.

= ((-m,t),[t,+m));

By ( i v ) Cy'cH,;

thus x'EH,,

i n j e c t i v e ( i i i ) ; thus (a) is false. that t

< ta.

Since t

which i s a b s u r d s i n c e F , i s

Hence ( b ) t h e r e e x i s t s sane a

By d e f l n i t i o n ( 3 , i ) , x l ( t )

~ ( y '

=

- ata(t),

and p(yl

-




+

0,

hB (1).

62

h(t)

Norman L. A l l i n g

>

0.

>

If t E T A then hA

TA; then tcTB#, hA

=

0 , and h A

0 , and hg

=

+

hg

>

1.65 Assume t h a t t i s n o t i n

0.

Thus hA + hg

h.

>

0.

Hence we see t h a t

C , a n d t h e ( l e x i c o g r a p h i c a l l y ) o r d e r e d d i r e c t sum A + B ,

are order-

i s a n o r p h i c ; t h u s t h e Theorem i s proved. A w i l l be c a l l e d

1.66

the c a n o n i c a l

d i r e c t summand of B i n G .

COMPLETE AND INCOMPLETE ORDERED GROUPS

Let C be an o r d e r e d group. EXAMPLE. (0)

(Z,+)

(a,+) a r e

and

complete, o r d e r e d group.

L e t C be a complete, Archimedean, o r d e r e d group; t h e n G is i s a n o r p h i c

t o one and o n l y one of t h e f o l l o w i n g : {O}, (Z,+) o r (R,+). PROOF.

U s i n g H B l d e r ' s Theorem ( 1 . 6 0 1 , we know t h a t G i s o r d e r -

i s o m o r p h i c t o a s u b g r o u p of (R,+). I G I = 1 , G = (01.

Let us i d e n t i f y t h e s e two groups.

I f G has a l e a s t p o s i t i v e element n , t h e n G

hence G i s order-isomorphic t o (Z,+).

no l e a s t p o s i t i v e element; then

C is

Assume t h a t

IGl

>

=

If

Z - n ; and

1 and t h a t C has

S i n c e C is c o m p l e t e , G

dense i n R .

=

(R,+); e s t a b l i s h i n g ( 0 ) . LEMMA.

Assume t h a t G is a m u l t i p l i c a t i v e o r d e r e d g r o u p ( w h i c h n e e d

not be A b e l i a n ) .

Then t h e f o l l o w i n g h o l d s :

(i)

i f G is non-Archimedean t h a n i t i s incomplete; t h u s

(ii)

i f G is complete i t i s Archimedean. PROOF.

Then t h e r e e x i s t s b

Assume t h a t t h a t G i s non-Archimedean.

a > 1 i n G such t h a t b n such t h a t g 4 a 1.

>

Let R

a", f o r a l l ncN. =

t h e union of L and R is C .

{gEC: g > a

n

, for

Let L

=

{gcG: t h e r e e x i s t s ncN

all mN}.

EL and bcR; t h u s C

=

Clearly L




Let

Let c be a c u t p o i n t

of C ; t h e n , b y d e f i n i t i o n , c i s e i t h e r ( a ) t h e g r e a t e s t element of L , o r ( B ) t h e l e a s t element of R .

For gcL, t h e r e e x i s t s ncN s u c h t h a t g 6 a n .

Preliminaries

1.66

Since a hold. h 5 a

n




0 and y

>

A w i l l be c a l l e d a n o r d e r e d

0 i m p l i e s xy

r i n g i s n e c e s s a r i l y a n i n t e g r a l domain.

>

0.

ring

Note t h a t an o r d e r e d

A f i e l d t h a t i s an ordered r i n g

w i l l be c a l l e d a n ordered f i e l d .

EXAMPLE.

2 is an o r d e r e d r i n g .

The f i e l d Q of r a t i o n a l numbers a n d

t h e f i e l d R of a l l r e a l numbers are o r d e r e d f i e l d s . Let A ( r e S p . F ) be an o r d e r e d i n t e g r a l domain ( r e s p . f i e l d ) , a n d l e t

P*

=

(x~A:x> 0).

( A s u s u a l , we d e f i n e A* t o b e A

-

(01.)

c a l l e d t h e s e t of s t r i c t l y p o s i t i v e elements of A ( r e s p . F ) .

P* w i l l be

Note t h a t P*

has t h e f o l l o w i n g p r o p e r t i e s :

(O*)

(i)

0LP*;

( i i ) f o r a l l XEA* then e i t h e r XEP* o r -xEP*; ( i i i ) P* is c l o s e d under a d d i t i o n ; and

(iv)

P* i s c l o s e d under m u l t i p l i c a t i o n .

I t i s a l s o c o n v e n i e n t t o d e f i n e P t o b e P * u n i o n (01.

c a l l e d t h e set of p o s i t i v e e l e m e n t s of A.

Let P b e

Then we have t h e f o l l o w i n g :

Norman L. A l l i n g (i)

1.70

P + P is a s u b s e t of P ,

( i i ) P-P is a subset of P,

( i i i ) t h e u n i o n of P a n d

(iv)

-P i s A , a n d

t h e i n t e r s e c t i o n of P a n d -P i s (0).

Let A b e a i n t e g r a l d o m a i n , a n d l e t P* b e s s u b s e t of A s a t i s f y i n g (Ox).

Let u s d e f i n e x


y z ; e s t a b l i s h i n g ( i v ) . ( v ) can b e r e s o l v e d b y t r e a t i n g PROOF.

=

1

=

t h e s e v e r a l cases s e p a r a t e l y .

(3)

o

For a l l X E A , ( i ) x 2 h 0 , a n d ( i i ) i f x f 0 , x z

>

0.

65

P r e l iminari es

1.70 if x L 0 then x2

PROOF.

L

I f x 5 0 t h e n -x 2 0.

0.

Thus x 2

2 0 ; e s t a b l i s h i n g ( i ) . To p r o v e ( i i ) , assume t h a t x 6 0 . i n t e g r a l domain, x 2 # 0.

(-x)~

=

S i n c e A i s an

Using t h i s f a c t and ( i ) , p r o v e s t h a t x 2

>

0.

a

=

a

Using ( 3 ) and ( O * , ( i i i ) ) , we s e e t h a t we have t h e f o l l o w i n g :

(4)

... ancA,

Given a l ,

such t h a t

n Ii=, a.’ 1

=

0, then a l

An i n t e g r a l domain B w i l l be c a l l e d f o r m a l l y

=

real i f

...

=

n

0.

( 4 ) h o l d s ; hence

a l l o r d e r e d i n t e g r a l domains are f o r m a l l y r e a l . (5)

( i ) B i s f o r m a l l y r e a l i f and o n l y i f

i s n o t a sum of s q u a r e s i n B.

( i i ) -1

Assume t h a t B is not f o r m a l l y r e a l ,

PROOF.

and b , ,

... bncB*,

n o t e t h a t -1

=

such t h a t

1.i =n 2 c i ’; t h u s

I,,,n

bi2

=

0.

Let c .

1

Then t h e r e e x i s t n =

>

1,

b . / b l , f o r a l l i , and 1

n o t ( i ) impiies not ( i i ) . Hence ( i i ) i m p l i e s

Now assume t h a t not ( i i ) h o l d s ; t h u s t h e r e e x i s t m 2 1 a n d d . i n B J m m d j 2 = 0 ; thus not ( i ) holds. d.’. Hence 1 ’ + such t h a t -1 = (i).

J

I,=,

Hence ( i ) i m p l i e s ( i i ) . S u p p o s e , f o r a moment, t h a t a f o r m a l l y r e a l domain were of c h a r a c t e r i s t i c p , f o r s a n e prime number p; t h e n 0 1 f 0.

=

lip1 1

=

1.’ 1’. 1=1

However,

Thus we c o n c l u d e t h a t t h e c h a r a c t e r i s t i c of e v e r y f o r m a l l y r e a l

domain i s 0. L e t A be a n o r d e r e d i n t e g r a l d o m a i n .

Let F be i t s f i e l d of q u o t i e n t s .

t e g r a l domain. a , b d , with b

(6)

>

0 , such t h a t f = a / b .

A s n o t e d a b o v e , A is a n i n Given f c F * , t h e r e e x i s t

D e f i n e f t o be p o s i t i v e i f a

T h i s d e f i n i t i o n of o r d e r on F i s i n d e p e n d e n t of r e p r e s e n t a t i o n .

>

0.

66

Norman L . A l l i n g

PROOF.

Let a / b

ab'

a'b.

Thus, a

(7)

Let P*(F)

>

=

f = a'/b',

w i t h a , b , a ' , b'cA

0 i f a n d o n l y i f a'

ifsF: f

=

>

1.70

>

0.

and b , b '

>

Let a , b , c a n d d be i n P*, l e t f =

ac/bd.

Then

0 ) ; t h e n P*(F) s a t i s f i e s ( 0 " ) .

S i n c e P * s a t i s f i e s ( O * , ( i ) & ( i i ) ) , P*(F) s a t i s f i e s ( O * , ( i )

bc)/bd and f g

0.

o

=

a/b, and l e t g = c/d.

S i n c e P* s a t i s f i e s ( O w ,

f

+

& (ii)).

g

=

(ad

+

( i i i ) & ( i v ) ) , s o does

P*(F); e s t a b l i s h i n g ( 7 ) .

(8)

P * ( F ) endows F w i t h t h e o n l y o r d e r under which F is a n o r d e r e d f i e l d whose o r d e r i n d u c e s t h e o r d e r g i v e n by P* on A.

l e t P * ' ( F ) be a s u b s e t of F t h a t s a t i s f i e s ( O * ) and t h a t

PROOF.

contains P*. Let f e F , and l e t f = a / b , w i t h a , b e A , a n d b f 0 . W i t h o u t l o s s o f g e n e r a l i t y we may assume t h a t b > 0. Assume t h a t f c P * ' ( F ) . S i n c e beP*, which is c o n t a i n e d i n P * * ( F ) , and s i n c e a Conversely, l e t fEP*(F).

Let F+ d e n o t e {xEF: x

Note t h a t F 1.71

+

f b , acP*.

Hence f E P * ( F ) .

If f i s n o t i n P * I ( F ) t h e n - f c P * ' ( F ) .

j u s t seen t h i s i m p l i e s t h a t -acP*;

(9)

=

>

which i s a b s u r d .

A s we h a v e

Thus f e P * ' ( F ) .

01.

is a s u b g r o u p of F* of i n d e x 2 .

THE ARTIN-SCHREIER THEORY OF REAL-CLOSED FIELDS

Let F be a f i e l d .

Let S ( F ) , o r S f o r s h o r t , b e t h e s u b c l a s s o f F

t h a t c o n s i s t s of 0 and a l l sums of s q u a r e s of elements of F*.

Then we have

t h e following:

(0)

(i)

oes;

is c l o s e d under a d d i t i o n a n d m u l t i p l i c a t i o n ; ( i i i ) F is f o r m a l l y r e a l i f and o n l y i f -1 is not i n S ; and ( i v ) f o r all a&* ( - S - [ O ] ) , l / a i s i n S*. (ill

o

S

P r e l i m i n a r i es

1 .71

( i ) is t r u e by d e f i n i t i o n .

PROOF.

be i n S .

Then a

Let a

b i s c l e a r l y i n S, as i s a.b

+

l i s h i n g ( i i ) . ( i i i ) follows frcm (1.70:4). then l / a

=

67

=

=

m lj=, aj2

m

n

1j=lI.‘k=l

and b

=

n 1k=l

b

k

( a . - b k ) * ; estabJ

A s t o ( i v ) , assume t h a t a # 0;

m ljSl (aj/a)2~S.

a(l/a)‘ =

Let F be an ordered f i e l d , and l e t P* be t h e s e t of a l l i t s p o s i t i v e elements.

EXAMPLE 0.

if F

Q then

=

i n Section 1.70, S* is a subset of P*.

As remarked

S*

If F

=

R, t h e r e a l number f i e l d , t h e n S*

=

P*.

However

i s a proper subset of P”.

A f i e l d F i s c a l l e d r e a l - c l o s e d i f F is formally r e a l and i f

i t has

no proper a l g e b r a i c extensions t h a t a r e formally r e a l . EXAMPLE 1 .

C l e a r l y t h e f i e l d of a l l r e a l numbers R i s a f o r m a l l y

We know t h a t t h e only a l g e b r a i c extension of R i s C , t h e f i e l d

real f i e l d .

of a l l complex numbers.

Since -1

=

i 2 , we s e e t h a t C is not formally r e a l ;

t h u s R i s real-closed. THEOREM 0.

Every e l e m e n t i n F* is

Assume t h a t F i s r e a l - c l o s e d .

e i t h e r a square o r i s t h e negative of a square. Since F is formally r e a l t h e r e e x i s t s CEF ( e . g . , - 1 )

PROOF.

n o t a square. that Y2

=

Let K be t h e s p l i t t i n g f i e l d of X 2

c ; thus K

formally r e a l .

F(Y).

=

-

c over F .

that is

Let YEK such

Since F is assumed t o be r e a l - c l o s e d , K i s not

T h u s t h e r e e x i s t n elements a and b . i n F , not a l l zero, j J

such t h a t

(1)

(i) (ii)

J=1

n

(a. J (a.’ J

b:Y)’ J

+

+

b.2.c) J

0: i . e . ,

=

=

-2.1

n j=1

( a . .b. ) .Y J J

Since Y i s not i n F we s e e from ( 1 , i i ) t h a t

2.1j=1n

( a .b j

j

) = 0 ; hence

68

Norman L. A l l i n g

n

a,'

+

J

n

c.1. b.* J=1 J

1 .71

0.

=

Since F is formally r e a l ,

(3)

n

bj2 f 0.

Assume f o r a moment t h a t t h e e x p r e s s i o n i n ( 3 ) i s 0.

PROOF.

is f o r m a l l y r e a l , each b . formally r e a l each a

j

=

Since ( 2 ) holds,

0.

J

0.

1.J =n1

a

J

*

= 0.

Since F

S i n c e F is

However, t h i s v i o l a t e s t h e c o n d i t i o n t h a t n o t

a l l a . and b . a r e z e r o . J J

(ii)

-cES,

( i i i ) c t S , hence (iv)

CES i m p l i e s t h a t

PROOF.

( 2 ) and

c is a square.

( 3 ) imply ( i ) .

By ( O , ( i v ) & ( i i ) ) ,-ceS; e s t a b -

l i s h i n g ( i i ) . Were CES t h e n by ( 0 , i v ) l / c would be i n S .

S i n c e -cES,

see t h a t C E S i m p l i e s - l e S , which i s absurd; e s t a b l i s h i n g ( i i i ) . have proved t h e f o l l o w i n g : t r a p o s i t i v e of

(A)

(A) C E F n o t a s q u a r e i m p l i e s c L S .

which i s t h e f o l l o w i n g :

we

Thus we The c o n -

(B) CCS i m p l i e s c i s a s q u a r e ;

establishing (iv). As t o t h e s t a t e m e n t of Theorem 0 , i f c is

-c&.

not a s q u a r e t h e n by ( 4 , i i )

By ( 4 , ( i i ) & ( i v ) ) , - c i s a s q u a r e ; t h u s c i s t h e n e g a t i v e o f a

s q u a r e ; proving Theorem 0. THEOREM 1 .

A r e a l - c l o s e d f i e l d F may be o r d e r e d i n one and o n l y o n e

way, namely w i t h t h e o r d e r g i v e n by P* = { x z : X E F * } .

F u r t h e r , any a u t a n o r -

phism of F is o r d e r - p r e s e r v i n g . PROOF.

Let P * b e d e f i n e d t o be { x ' :

(1.70:0(i)) holds. ( 1 . 7 0 : 0 ( i i ) ) holds.

XEF*}.

C l e a r l y OtP*;

thus

Let C E F - P*; t h e n by Theorem 0 , - c i s i n P*; t h u s

Let a , bEF*; t h e n a 2 - b 2

= (ab)2,

we s e e t h a t a'eb'

is

1 .71

Preliminaries

i n P*, hence ( 1 . 7 0 : 0 ( i v ) ) h o l d s .

Were a '

69 + b 2 n o t i n P * t h e n we would

know, b y Theorem 0 . t h a t i t was - c 2 , f o r s a n e C E F ; t h u s a' Since F is formally real t h i s i m p l i e s t h a t a

=

b = c

=

b2

+

+

p o s i t i v e e l e m e n t s of F .

=

0.

0 ; which i s a b s u r d .

Thus P* is c l o s e d under a d d i t i o n , showing t h a t ( 1 . 7 0 : 0 ( i i i ) ) h o l d s . we know t h a t (1.70:O) h o l d s .

c2

Hence,

As a r e s u l t P * may be t a k e n a s a c l a s s of

S i n c e any s e t of p o s i t i v e e l e m e n t s of F must

c o n t a i n t h e n o n - z e r o s q u a r e s ( 1 . 7 0 : 3 ) , we s e e t h a t t h e o r d e r on F i s unique. P*,

Let h be an automorphism of F.

Since h preserves squares h(P*)

=

t h u s h i s o r d e r - p r e s e r v i n g ; proving Theorem 1 . Henceforth i n t h i s S e c t i o n assume t h a t a l l f i e l d s under c o n s i d e r a t i o n

are sets. Let A be a f o r m a l l y r e a l f i e l d and l e t C be an a l g e b r a i c

THEOREM 2.

c l o s u r e of A .

There e x i s t s a r e a l - c l o s e d f i e l d B t h a t i s a s u b f i e l d o f C

and t h a t c o n t a i n s A . PROOF.

contain A.

L e t E be t h e s e t of a l l f o r m a l l y r e a l s u b f i e l d s of C t h a t

Since A i s f o r m a l l y r e a l A C E , t h u s E f 0 . Let

r

t h e u n i o n F of

r

inclusion.

Let E be o r d e r e d by

b e a non-empty ( t o t a l l y ) o r d e r e d s u b s e t of E.

i s a g a i n i n E ; thus E is inductive.

has a maximal e l e m e n t , B.

Clearly

By Zorn's Lemma, Z

By c o n s t r u c t i o n , B i s r e a l - c l o s e d , p r o v i n g

Theorem 2. (5)

Let A be a f o r m a l l y r e a l f i e l d ; t h e n A c a n be embedded i n a r e a l c l o s e d f i e l d B such t h a t B is a l g e b r a i c over A . PROOF.

(6)

Apply Theorem 2 .

o

I f A i s f o r m a l l y r e a l , t h e n i t can be o r d e r e d . PROOF.

Apply ( 5 ) .

S i n c e B is r e a l - c l o s e d , we may a p p l y Theorem 1

and t h u s we know t h a t B has a unique o r d e r on i t , g i v e n by P*

=

{x':

XCB*).

Let P*, be t h e i n t e r s e c t i o n of P* and A ; t h e n P*, s a t i s f i e s ( 1 . 7 0 : 0 ) , a n d t h u s B can be o r d e r e d by P*,.

a

Norman L. A l l i n g

70 THEOREM 3.

degree.

1.71

Let F be a r e a l - c l o s e d f i e l d .

Let f ( X ) i n F[X] be of odd

Then f ( X ) has a r o o t p i n F .

PROOF.

Let n be t h e d e g r e e of f ( X ) .

I f n = 1 then c l e a r l y f ( X ) h a s

Assum e t h a t n i s an odd number g r e a t e r t h a n 1 f o r which a l l

a root i n F.

elements i n FCX] of odd d e g r e e l e s s t h a n n h a v e r o o t s i n F .

Were f ( X )

r e d u c i b l e t h a n i t would f a c t o r i n t o two polynomials a(X) and b(X) of lower degree i n F[X].

Since n i s odd, t h e d e g r e e of a(X) o r b ( X ) i s o d d .

t h a t p o l y n o m i a l has a r o o t i n F .

generality, that f ( X ) is irreducible.

Thus

H e n c e , we may assume, without loss of Let L b e a f i e l d e x t e n s i o n of F such

t h a t f ( X ) has a r o o t p i n L , f o r w h i c h L

=

S i n c e L i s a proper

K(p).

a l g e b r a i c e x t e n s i o n of F , a r e a l - c l o s e d f i e l d , L is n o t f o r m a l l y r e a l . Thus t h e r e exist c . E L , J

c

j

... , m ,

for j = 1 ,

with

m lj=, cj

S i n c e each

= -1.

i s i n L we know t h a t f o r e a c h t h e r e is a p o l y n o m i a l p (X)eF[X], of

J

d e g r e e l e s s t h a n n . such t h a t p ( p ) j

=

c.. J

Thus, t h e r e e x i s t s a g ( x ) ~ F [ X l

such t h a t t h e f o l l o w i n g h o l d s :

m

=

-1

f(X)g(X

i-

I t i s e a s i l y s e e n t h a t t h e l e a d i n g c o e f f i c i e n t of i s a s u m of s q u a r e s i n F , a n d hence i s p o s i t i v e .

s ( X ) i s even and i s bounded above by 2 ( n

-

1).

1J = 1

p (XI2

J

=

s(X),

F u r t h e r , t h e degree of

It follows t h a t t h e d e g r e e

of g ( X ) i s odd and i s bounded above by 2 ( n - 1 ) - n

=

n

-

2.

t h e i n d u c t i o n h y p o t h e s i s , we know t h a t g ( X ) h a s a r o o t B E F .

On invoking

Hence ( 7 )

gives rise t o

B u t ( 8 ) i s a b s u r d s i n c e F i s a r e a l - c l o s e d f i e l d and hence i s a f o r m a l l y

real f i e l d .

T h u s f ( X ) has a r o o t p i n F ; proving Theorem

THEOREM 4 .

3.

Let F be an o r d e r e d f i e l d s u c h t h a t ( i ) every p o s i t i v e

element i n F is a s q u a r e and ( i i ) every p o l y n o m i a l of odd d e g r e e i n F [ X l

P r e l i m i nar i es

1 .71

has a r o o t i n F.

Then P ( X )

S P l t t i n g f i e l d of f ( X

=

X2

+

71

IEF[XI is i r r e d u c i b l e .

L e t C be t h e

over F ; t h e n C is a l g e b r a i c a l l y c l o s e d . A s we h a v e

S i n c e F is an o r d e r e d f i e l d i t i s f o r m a l l y r e a l .

PROOF.

s e e n , F h a s c h a r a c t e r i s t i c 0 ; t h u s C i s a normal s e p a r a b l e e x t e n s i o n of F. C l e a r l y i t s C a l o i s group Go i s t h e two element group.

of f ( X ) i n C .

L e t x ( p ) be defined t o be a - b i ; t h e n

a + bi.

Let k i be t h e r o o t s

Given ~ E C ,t h e r e e x i s t unique a and b i n F such t h a t p

Let f ( X ) E C ( X ) , by t a k i n g X t o X .

=

x and x 2 c o n s t i t u t e

Let t h e F-automorphism

x

Go.

of C e x t e n d t o

an F-autanorphism

x of

Since X(h(X))

h ( X ) , ~ ( X ) E F [ X ] . If h(X) h a s a r o o t i n C t h a n f ( X ) has a

root i n C .

(9)

=

Let g(X)

C[X].

=

X ( f ( X ) ) , and l e t h(X) = f ( X ) * g ( X ) .

Thus,

t o show t h a t C i s a l g e b r a i c a l l y c l o s e d i t s u f f i c e s t o show t h a t e v e r y polynomial w i t h c o e f f i c i e n t s i n F has a r o o t i n C . U s i n g c o n d i t i o n ( i i ) of Theorem 4, we know t h a t t h i s i s t r u e f o r a l l

pol ynom i a 1s of odd d e g r e e w i t h c o e f f i c i e n t s i n F .

(10)

Every element p i n C has a s q u a r e r o o t i n C .

PROOF r o o t i n F.

If p

>

0 t h e n , by c o n d i t i o n ( i ) i n Theorem 4 , p h a s a s q u a r e

Assume t h a t p

such t h a t B 2

=

are i n F , with b 6 0. 2cdi.

(11)





0 , then f ( s )

0 f o r a l l s c R , and.

( i i ) if a




0, let vx(q(X)) be t h e

L e t u s d e f i n e ~ ( 0 )=

1 , r is c a l l e d a s i m p l e zero of q ( X ) .

s i m p l e p o l e of q ( X ) .

an e l e m e n t

m,

0 , q ( X ) i s s a i d t o h a v e a z e r o of o r d e r

s a i d t o h a v e a pole of o r d e r - n a t A.

If n

=

-1,

If n




+ 0.

(0,O)

and g ( q ( x , y ) )

>

Norman L . A l l i n g

116

(3)

(q-'(T))*

is a €,-open

3.42

Thus q is 6-continuous on C*.

s u b s e t of C * .

( 1 ) and ( 3 ) being t h e c a s e , we w i l l say t h a t C a t o p o l o g i c a l f i e l d i n t h e 6-topology. -

3.43

OTHER EXAMPLES OF c-CONTINUOUS MAPS

Let u s c o n s i d e r a few examples of s p e c i a l maps fran R m t o R n . be a l i n e a r map form R (0)

(1)

t o Rn.

Then, by ( 3 . 2 1 : 0 ) ,

f i s a €,-continuous map. Let M n x m ( R )

R.

m

Let f

d e n o t e t h e s e t of a l l mxn m a t r i c e s w i t h c o e f f i c i e n t s i n

A s a c o r o l l a r y t o (3.21:O) we s e e t h a t

m

t h e map t h a t takes ( A , X ) E M ~ ~ ~ ( R )t X o RAXER

where h e r e we t h i n k of R

n

, is a c-continuous map,

m and R n as a s p a c e of column v e c t o r s .

Let G L ( n , R ) d e n o t e t h e g e n e r a l l i n e a r group, of n by n m a t r i c e s over

A.

(3)

A s another c o r o l l a r y t o ( 1 ) we see t h a t

n

t h e map t h a t takes (A,x)cGL(n,R)xR

t o AXER n is a 6-continuous map,

where h e r e we t h i n k of R n as t h e s p a c e of column v e c t o r s .

117

CHAPTER 4

I N T R O D U C T I O N T O THE SURREAL NUMBER F I E L D No

4.00

SURREAL NUMBERS

In J . H . Conway's book, On Numbers and Games C241, t h e b a s i c c o n s t r u c t i o n o f numbers i s t h e f o l l o w i n g : (0)

I f L a n d R a r e two s e t s of n u m b e r s , a n d i f no member of L is t any

member of R , t h e n ( L I R } i s a number.

A l l numbers are c o n s t r u c t e d i n

t h i s way [ 2 4 , p . 41.

How t h e n d o e s o n e g e t s t a r t e d c o n s t r u c t i n g n u m b e r s u s i n g C o n w a y ' s

construction?

The empty s e t is a s e t of numbers which we know e ists.

L and R be empty. (01,

Note t h a t no member of L is 2 any member of R

[LIR] i s a number.

Let u s c a l l t h i s number 0 .

Let

t h u s , by

Conway C24, p . 41

adopted t h e following n o t a t i o n a l convention:

If x

=

( L I R } w e w r i t e xL f o r a t y p i c a l member of L , a n d

t y p i c a l member of R ; t h u s x e, f , e, f ,

... ) , ... 1 .

option _---

of x.

we mean t h a t x

L R {x Ix 1 .

= =

If we write x = ( a , b , c ,

... I d ,

and R

=

Id,

x L i s c a l l e d a l e f t o p t i o n of x , a n d x R i s c a l l e d a r i g h t If L ( r e s p . R ) i s empty, we may i n d i c a t e t h i s by l e a v i n g t h e

p l a c e where L ( r e s p . R ) would a p p e a r b l a n k . =

... }

( L I R ) , where L = [ a , b , c ,

xR for a

Hence ( ( 0 1 l a }

=

([O}

I],

and 0

[I). I n K n u t h ' s m a t h e m a t i c a l n o v e l l a o n s u r r e a l n u m b e r s [52] he u s e s

s l i g h t l y d i f f e r e n t n o t a t i o n i n t h e body of t h e t e x t . writes x

=

(X

X 1. L' R

For example, Knuth

We have c h o s e n t o a d o p t most of Conway's n o t a t i o n .

is n o t o n l y v e r y compact a n d e a s y t o u s e , b u t i t s u g g e s t s

feels

-

t h e r i g h t way t o t h i n k a b o u t t h e s u b j e c t .

-

It

t h e author

118

Norman L . A l l i n g

4.00

Conway t h e n d e f i n e s o r d e r between numbers a s f o l l o w s :

(1)

( 1 ) x 6 y i f and O n l y i f ( i i ) no y R 2 x and x S no x

L

.

Note t h a t ( 1 , i ) is a s t a t e m e n t about n u m b e r s , a n d t h a t ( 1 , i i ) i s a s t a t e m e n t a b o u t s e t s of n u m b e r s .

Conway d e s c r i b e s 0 as t h e t l s i m p l e s t t t

number t h a t was ttborn on day 0" [24, p.

111.

T h i s seems f i t t i n g i n d e e d ,

{*I.]. The numbers 1 = Conway s a y s of them t h a t

s i n c e i t i s b u i l t up fran t h e empty s e t u s i n g o n l y

{Ol) and -1

-

(10) a r e a l i t t l e more c o m p l e x .

t h e y were e a c h " b o r n o n d a y 1 " [ 2 4 , verify t h a t ( 1 , i i ) holds. and t h a t ( b ) 0 < j l ) ,

p . 111.

To s e e t h a t 0 2 1, w e m u s t

To do t h a t i t s u f f i c e s t o show t h a t ( a ) lo)


x i f and o n l y i f x < y.

(ii)

(iii)

Perhaps t h e o n l y s u r p r i s e i s t h a t ( 2 , i i ) is a definition.

Conway

y, -x,

and xy

ends h i s s h o r t l i s t of remarkable s t a t e m e n t s by d e f i n i n g x

+

i n d u c t i v e l y f o r all numbers x and y as f o l l o w s . L R R I x L + y , x + y Ix + y , x + y 1 .

(3)

x + Y

(4)

-x = (-x

(5)

x y - ( x y + x y

=

R

L

1-x I .

L

L x Y

+

L

L L R R R R - x y , x y + x y - x y J R L R R L X Y - x y , x y + xy - x Ry L ) *

A t f i r s t g l a n c e t h e s e d e f i n i t i o n s may l o o k c i r c u l a r .

Note, f o r

example, i n ( 4 ) i f we know how t o form t h e n e g a t i v e of a l l t h e o p t i o n s of x used t o d e f i n e x , t h e n (4) i s n o n - c i r c u l a r .

S i m i l a r l y , i n ( 3 ) i f we c a n

p r e f o r m a l l t h e i n d i c a t e d a d d i t i o n s among o p t i o n s of x and y and y and x

I n t r o d u c t i o n t o t h e s u r r e a l number f i e l d No

4.00

t h e n w e c a n compute t h e s e t s on t h e l e f t i n ( 3 ) .

119

The same may b e s a i d of

(5). Conway a l s o showed C24, p p . 16-17] t h a t , i f x

(6)

L


X I . S i n c e by d e f i n i t i o n t h e r e e x i s t s nEN s u c h t h a t -n < x < n , L and R a r e non-empty. Clearly

( i i i ) Let L = { q E Q : (0)

-

-

R x )(Y

-

(x

and y = { y

y , x + y + 2 - m ) ; showing t h a t x + y i s

{xy - ( x {xy

( i i ) Let x a n d y b e r e a l

Using (4.08:19) we know t h a t xy =

L L x ) ( y - y 1,

-

L

-

xy

(x

2-"1

may be w r i t t e n as

a n d t h u s - x i s a r e a l number i n No.

a r e a l number i n No. {xy

-

(x

=

2-"),

+

dED

By ( 4 . 0 9 : 1 ) , x i s r e a l .

n u m b e r s i n No, w i t h x

(X

4.30

L and {x

coinitial. (4.02:16),

-

2-"]

q

a r e m u t u a l l y c o f i n a l a n d R and { x + 2-"]

By ( 4 . 0 2 : 1 6 ) , x { L I R ) is real.

As we have

=

(LIR].

are mutually

( i v ) Let ( L , R ) b e a g a p i n Q .

o

see i n S e c t i o n 4.21, 0

w

i s t h e r i n g D of d y a d i c n u m b e r s .

- D

S i n c e D i s d e n s e i n t h e f i e l d of real numbers R , a number r i n R associated w i t h s u b s e t s L

Clearly L < R.

=

{acD: a

< r)

a n d R = {bED: b

>

i s a t i m e l y r e p r e s e n t a t i o n of x.

Let x

=

= w.

c a n be

r ] of O w .

Let x = { L I R ] , a n d n o t e t h a t x i s n o t i n 0

{No, c

9,.

W i t h o u t l o s s of g e n e r a l i t y we may assume t h a t ( i )

f o r a l l SES.

sor of S, s^(B) =

There e x i s t s o ,

B)

Thus s

=

+,

Were t h e r e

8,hS

f o r a l l SES.

w i t h s,^(B) = 0 , s,

Hence

BEr,

which i s

absurd.

PROOF. so S c 6

bt(c)


x and b(y)
a , x is not i n

is a Cuesta Dutari c u t i n F ( < , a ) .

{La(x)l Ra(x)}, and n o t e that x a c F ( = , a ) .

Let xu

S i n c e B > a , x f x a'

=

Recall

(4.50) t h a t Conway C24, p.291 c a l l s xa t h e u t h a p p r o x i m a t i o n t o x.

Let us c a l l yeF a p r e d e c e s s o r ( c f . ( 4 . 5 0 ) ) of x , a n d write y

( c f . ( 4 . 5 1 ) ) i f there e x i s t s a




< 8.

Note t h a t x ( a ) =

+

(resp. -1 i f f x

(resp. y




xu ( r e s p . x

thus y[B = x.

The f o l l o w i n g w i l l be c a l l e d t h e p r e d e c e s s o r

< x and a


x and a

x

iff

Since b ' ( y ) 5

cut r e p r e s e n t a t i o n

of x:

< 61).

(1)

((xu: xa

(2)

Let (L,R) be t h e p r e d e c e s s o r c u t r e p r e s e n t a t i o n of x.

a




0, t h e r e e x i s t a

, using




0.

Following t h e same a r g u m e n t as t h a t u s e d

t o prove Lemma 1 , i n S e c t i o n 4.40,

e x i s t s a unique yL ( r e s p . y R L R w i t h u Y ( r e s p . wY

=

we see t h a t f o r each x

L

R

(resp. x ) there

L

i n CNo s u c h t h a t xL

R

( r e s p . xR

a W'

a

u Y 1,

R

a t l e a s t as s i m p l e as xL ( r e s p . x 1.

I f x i s commens u r a t e w i t h one of its o p t i o n s , s a y x ' , t h e n ( i i i ) i s p r o v e d , i n t h e c a s e

under consideration.

Assume t h a t x i s c o m m e n s u r a t e w i t h n o n e of i t s

rwYL XI.

Given X E O , l e t L ( x ) =

S i n c e x i s i n 0, ( L ( x ) , R ( x ) ) is

S i n c e t h e f i e l d R i s Dedekind c o m p l e t e , t h e r e exists

a unique c u t p o i n t p ( x ) i n R f o r ( L ( x ) , R ( x ) ) .

Then o n e e a s i l y sees t h a t

(0)

p is a p l a c e of No a s s o c i a t e d w i t h 0 .

(1)

R is a s u b f i e l d of 0 t h a t p maps R - i s o m o r p h i c a l l y o n t o t h e r e s i d u e c l a s s f i e l d of p .

Norman L. A l l i n g

21 2

6.10

Let 5 b e a p o s i t i v e r e g u l a r index (1.30:3).

R e c a l l t h a t (No ( 5 . 0 0 )

R e c a l l a l s o t h a t R is a s u b f i e l d of CNo

i s a s u b f i e l d o f No ( 5 . 0 0 ) .

Let < p d e n o t e plFNo, a n d l e t 50 d e n o t e 0 i n t e r s e c t e d w i t h gNo;

(5.00).

then R i s a s u b f i e l d of 50 t h a t c p maps R - i s a n o r p h i c a l l y o n t o t h e r e s i d u e

(2)

class f i e l d R of (p.

I f i t i s u n l i k e l y t h a t c o n f u s l o n w i l l a r i s e we may use p t o d e n o t e s p and use 0 t o d e n o t e 50.

R e c a l l t h a t t h e w-map was d e f i n e d on No i n ( 4 . 4 0 ) .

According t o Lemma 2 of S e c t i o n 4.50. f o r a l l y i n No t h e r e e x i s t s a unique XENO such t h a t y

a

w

-X

,

where

a

d e n o t e s t h e e q u i v a l e n c e r e l a t i o n o n No

between commensurate e l e m e n t s ( 4 . 1 0 ) . element XENO s u c h t h a t I y I

(3)

a

w

-X

For a l l YENO*, l e t V ( y ) b e t h e

.

V i s a homomorphism of t h e m u l t i p l i c a t i v e group No* o n t o t h e

(i)

a d d i t i v e group (No,+) of No. ( i i ) The k e r n e l of V is U.

( i i i ) For a l l y and Y'ENo*, l y l

v(o)

(iv)

PROOF.

holds.

By Theorem 4.40, =

-X

wX+'

we s e e t h a t V - l ( O )

=

>

V(y*).

Finally,

w X w y , f o r a l l x and y i n No, t h u s ( i )

IyI = w

0 iff

f r a n Lemma 1 of S e c t i o n 4 . 4 0 . I,I

V(y)

= NO+.

For ycNo, V ( y )

(6.00:4),

.

xi).(^,,,

( bj ) x j ) , w h i c h by

By t h e Lemma ab o v e we see t h a t

1 ; t h e n ( ( 1 + x ) 1'k)k

=

1 + x.

We w i l l c a l l a n o r d e r e d f i e l d K a r o o t - c l o s e d f i e l d i f f o r e a c h k s N and each a

>

0 i n K t h e r e e x i s t s bsK s u c h t h a t b

f i e l d , l e t F have t h e l e x i c o g r a p h i c o r d e r .

we w i l l p u t on F , e a c h t g COROLLARY 2.

>

k

=

a.

I f K is a n o r d e r e d

Under t h i s o r d e r , t h e o n l y o n e

0.

Assume t h a t K i s a r o o t - c l o s e d f i e l d , a n d t h a t C i s

Power s e r i e s : formal and hyper-convergent

7.33

275

d i v i s i b l e ; then F is r o o t - c l o s e d .

PROOF.

r i s i n K and i s p o s i t i v e .

=

r-'at-g

c l o s e d , t h e r e e x i s t s scK such t h a t s b

k

=a.

Let V(a)

Let aEF be p o s i t i v e and l e t kEN.

1 + x , where XEM.

=

k

=

r.

Let b

=

=

g ; then p ( a t - g ) Since K is r o o t -

stgIk.(l

+

x)'Ik;

then

~3

Combining t h e s e r e s u l t s we see t h a t we have proved Conway's Theorem 24 C24, p . 401, namely t h e f o l l o w i n g . COROLLARY 3 .

Every p o s i t i v e a i n No h a s a u n i q u e n-th r o o t , f o r

every p o s i t i v e i n t e g e r n. T h e r e e x i s t s a p o s i t i v e r e g u l a r index gcOn, s u c h t h a t accNo.

PROOF.

t h e r e i s a n a t u r a l R-isomorphism

By T h e o r e m 6 . 5 5 , gR(((No,+))). root-closed.

f of

By C o r o l l a r y 2 , @ ( ( ( N o , + ) ) ) i s r o o t - c l o s e d ;

gNo o n t o t h u s gNo is

S i n c e F is an o r d e r e d f i e l d , c i s u n i q u e , e s t a b l i s h i n g t h e

C o r o l l a r y , and hence Conway's Theorem 24, i n t h e way t h a t h e s u g g e s t s . POWERS AND VALUES OF NEUMANN SERIES

7.34

F o r t h e moment l e t us drop t h e assumption t h a t t h e c h a r a c t e r i s t i c of K is n e c e s s a r i l y 0.

Let ( a n ) n L Obe a s e q u e n c e i n K , a n d c o n s i d e r t h e

f o l l o w i n g Neumann series:

(0)

~ ( x =)

In,, W

anxn , f o r each x c ~ .

By Neumann's Theorem (7.211, A(x) i s an element of 0, t h e v a l u a t i o n r i n g of K ( ( x ) ) .

Thus XEM + A(x)EO i s a w e l l - d e f i n e d mapping f r a n tl i n t o 0 ,

w h i c h we w i l l d e n o t e by A .

~ ' ( 1 anx ~ :n-~1 ) ,

(7.31:1,v).

we see t h a t (1)

xcH

+

A(X)EM + a,.

Assume t h a t a, Since

I,,:,

-

0.

We know t h a t A ( x ) =

anx n- 1 is an element i n 0 (7.301,

276

Norman L. A l l i n g

7.34

Now l e t x b e any non-zero element i n F.

Let S d e n o t e t h e s u p p o r t o f

x ; t h e n S i s a n o n - e m p t y , w e l l - o r d e r e d s u b s e t of G .

Let g o be t h e l e a s t

element of S.

(2)

n For a l l XEF*, t h e l e a s t element of s u p p ( x ) i s n - g o , f o r a l l nsN. C l e a r l y t h e s t a t w e n t b e f o r e t h e s e c o n d comma i n ( 2 ) is t r u e

PROOF.

for n

=

Let i t be t r u e f o r s a n e neN.

1.

tained i n supp(x)

+

W e know t h a t supp(xn+'

i s con-

n

supp(x ) ( 6 . 2 0 : 5 ) , whose l e a s t e l e m e n t i s ( n + l ) . g , .

By d e f i n i t i o n (6.201, x n + ' ( ( n

+

-

l).g,)

X"(n*g,)*X(g,) b 0.

0

L e t V be t h e Hahn v a l u a t i o n o n F ( 6 . 2 0 ) ; t h e n

(3)

n

V(x ) = ng,,

(i)

( i i ) For XEM

-

for all

nEZ.

[ O J , V(A(x))

such t h a t a

n

=

ng, = n - V ( x ) , where nEZ(L0) is minimal

f 0.

f , t h e K-monmorphism d e f i n e d i n Theorem

For each keN, A ( X ) k =

In:o

7.30, m a p s A(X) t o A ( x ) E F .

an,kXn, where t h e a

n ,k

are i n K .

(4)

( A ( x ) ) ~ is an element i n 0 of t h e f o l l o w i n g form:

(5)

I f a, f 0 , t h e n XEH

PROOF. an(xl

-

A(x,))

=

n

+

A(x)EH

+

Thus,

n an,kx

a, is a n i n j e c t i o n .

Let x, and x, be d i s t i n c t e l e m e n t s i n M; t h e n A ( x , )

- x,").

V ( x , ) fi

Assume ( i ) t h a t x, = 0 ; t h e n x , f 0. m;

t h u s A(x,) f A ( x , ) .

-

-

A(x,)

=

By ( 3 1 , V(A(x,)

Assume ( i i ) t h a t x, f 0 f x , ;

t h e n u s i n g ( 3 1 , we know t h a t V ( A ( x , ) - A ( x , ) ) = V(x, f 0 , we s e e t h a t V(A(x,)

.

A ( x , ) ) b -, and hence A ( x , )

-

xo). f

S i n c e x,

A(x,).

-

xo

7.35

27 7

Power s e r i e s : formal and hyper-convergent

7.35

COMPOSITION OF NEUMANN SERIES

Let

and (bn)neZ(20)

(am)mEN

be sequences i n K , and l e t t h e f o l l o w i n g

be d e f i n e d :

Let W denote t h e Hahn v a l u a t i o n of K ( ( X ) )

W(a)

=

0, f o r a l l

n

aEK*.

(6.20); t h e n W ( X )

= 1,

and

Note t h a t W ( A ( X ) ) 2 1 ; t h u s

) l n E Nis a s t r i c t l y i n c r e a s i n g sequence i n N .

(1)

(W(X

(2)

Assume, f o r a moment, t h a t bn

t h e n B ( X ) is a polynomial i n X .

=

>

0, for a l l n

k;

C l e a r l y t h e r e i s no d i f f i c u l t y i n d e f i n i n g

B ( A ( x ) ) , e s t a b l i s h i n g t h a t i t i s an element C ( X ) E K [ [ X ] ] ,

and t h a t B(A(x))

=

C ( x ) , f o r a l l XEM. Now l e t u s d r o p a s s u m p t i o n ( 2 ) . element C ( X )

ljmo cjXJ

=

i n K[[X]]

I s t h e r e any hope of d e f i n i n g a n

t h a t i s , i n some s e n s e , " B ( A ( X ) ) " ?

S i n c e ( 1 ) h o l d s , t h e o n l y powers of A ( X ) t h a t may c o n t a i n non-zero terms of t h e form c X J ,

f o r sane CEK, a r e t h e following: A ( X )

0

,

A(X)

1

,

...

,

A(X)J.

Thus we s e e t h a t

(3)

expanding

lnIobn(l,z,

t h e form c X J ,

LEMMA.

PROOF.

g i v e s r i s e t o an element C ( X )

For a l l x i n M, B ( A ( x ) )

Let x be i n M.

Recall t h a t f o r gEC InEN:

mn amX ) f o r m a l l y , and adding t o g e t h e r terms of

gEn-S) ( 7 . 2 2 ) .

-

was,

Thus

=

=

1." J=o

C.XJEK"XI]. J

C(x).

S = s u p p ( x ) i s a w e l l - o r d e r e d s u b s e t of .'C

m(g) = 0 , and f o r gcw.S, m(g) = 1 + max.

Norman L . A l l i n g

278

7.35

We have s e e n ( 7 . 2 2 ) t h a t s u p p ( A ( x ) ) , which we w i l l d e f i n e t o be T , i s

a s u b s e t of t h e w e l l - o r d e r e d s e t w - S of G'. a n d f o r gew*T, l e t n ( g ) = 1 + rnax.{neN: B(A(x))(g) =

)1 ;:

For g E ( G gEn.T).

- w*T),

l e t n(g) = 0,

Then, by d e f i n i t i o n ,

bn(A(xIn(g)) (7.22:2).

Fran (4) we see t h a t

On expanding t h e r i g h t hand s i d e of (51, a d d i n g a l l terms of t h e form c x J ( g ) , and r e c a l l i n g ( 3 1 , we see t h a t B(A(x)) = C ( x ) .

7.36

THE EXPONENTIAL SERIES AND THE LOGARITHMIC SERIES

Assume t h a t t h e c h a r a c t e r i s t i c of t h e f i e l d K is 0.

Let x b e i n M,

and c o n s i d e r t h e f o l l o w i n g d e f i n i t i o n :

By Neunann's Theoren we know t h a t e x p is w e l l - d e f i n e d on M a n d maps M

i n t o 0. We w i l l c a l l t h e Neumann series o n t h e r i g h t i n (0) t h e exponent i a l series. We w i l l c a l l exp t h e e x p o n e n t i a l f u n c t i o n . C l e a r l y

-(1)

t h e e x p o n e n t i a l f u n c t i o n maps M i n t o 1 + M.

PROOF.

Let x and y be i n M; t h e n , by ( i ' . 3 1 : 1 ) ,

exp(x)*exp(y) =

7.36

Power s e r i e s : f o r m a l a nd h y p e r-c o n v e rg e n t

A companion

27 9

series t o t h e exponential s e r i e s i s t h e logarithmic

s e r i e s , namely t h e f o l l o w i n g Neumann series: l e t x b e i n M and d e f i n e

we know t h a t t h e series o n t h e r i g h t of F u r t h e r , i t t e l l s u s t h a t l o g maps 1 + n i n t o M.

By Neuman n's Theorem (7.221,

(2) i s hyper-conve r ge nt.

For a l l XEM t h e f o l l o w i n g h o l d : (i) l o g ( e x p ( x ) )

THEOREM 1 . ( i i ) exp(log(1 maps 1

+

+

x))

=

1 + x.

Thus, ( i i i ) e x p maps M o n t o 1

+

=

x, and

H, a n d l o g

U onto M.

PROOF.

Using Lemma 7.35 we know t h a t t h e c o e f f i c i e n t s of t h e Neumann

series f o r l o g ( e x p ( x ) ) , and e x p ( l o g ( 1 + X I ) , expanded i n powers of x c a n be computed by c o n s u l t i n g t h e c o m p o s i t i o n s of t h e c o r r e s p o n d i n g f o r m a l power

series.

That t h e s e f o r m a l power series w i t h r a t i o n a l c o e f f i c i e n t s h a v e t h e

r e q u i r e d p r o p e r t i e s f o l l o w s f r o m t h e f a c t t h a t t h e same power s e r i e s , r e g a r d e d a s c o n v e r g e n t power s e r i e s o v e r t h e c o m p l e x numbers, h a v e t h e required properties. THEOREM 2 .

PROOF.

M , and exp(x) =

Thus t h e r e q u i r e d i d e n t i t i e s i n Q must h o l d .

For a l l u a nd v i n 1

Let l o g ( u ) =

=

+

x and l o g ( v )

u , a n d exp(y)

=

log(exp(x).exp(y)) = log(exp(x

v. +

H, l o g ( u * v ) =

=

log(u)

+

log(v).

By Theorem 1 , x a n d y a r e i n

y.

Using Theorem 0, we know t h a t l o g ( u . v ) y))

=

x

+

y

=

log(u)

+

log(v).

Norman L . A l l i n g

280

7.40

FORMAL POWER S E R I E S R I N G S I N A F I N I T E NUMBER OF VARIABLES

7.40

L e t K be any f i e l d and l e t ncN.

Let V E Z ( > O ) ~ , be thought of as a

Throughout t h i s Section v w i l l be i n Z ( 2 0 ) n .

multi-index,

d e f i n e d t o be

lif=l,v i ~ Z ( > O ) .

Z ( L 0 ) " i n t o K:

t h u s i f A is a K-valued c o e f f i c i e n t , A ( v , ,

Let s u m ( v ) b e

Let a K-valued c o e f f i c i e n t be a map

...

,vn)

A

fran

=

A(v)

i n K . f o r a l l v€Z(LO). By a formal power s e r i e s i n n v a r i a b l e s w i t h coef-

ficients

(O)

2 K,

"sum(v)=k

'k10

where X

w i l l be meant t h e following k i n d of expression:

=

(Xl,

A(vl,

... , X n )

... ,vn)X1 v1 ... * X n

V

")

=

i s a v e c t o r of n i n d e t e r m i n a t e s .

( F o r a more

p r e c i s e d e f i n i t i o n , d e f i n e t h e map A t o be t h e f o r m a l power s e r i e s i n q u e s t i o n , and proceed i n t h e obvious way.) Let A ( X ) be such an expression (0).

Let K[[X,,

... ,Xn]],

or simply K[[X]],

s i o n s of the k i n d given i n (0). K[[X,,

denote t h e s e t of a l l expres-

... , X n ] ]

and K"Xl1

w i l l be c a l l e d

t h e r i n g of f o r m a l power s e r i e s i n n v a r i a b l e s and c o e f f i c i e n t s 1_;

K.

Under formally defined o p e r a t i o n s , KCCXl] is an i n t e g r a l domain, a s well as

being a vector space over K . Assume t h a t A(v) C 0 .

Then, A(v)XV i s s a i d t o be of degree v and order

I f swn(v) = 0 then t h e monomial i n q u e s t i o n w i l l be i d e n t i f i e d

sum(v).

w i t h t h e c o n s t a n t A(v) i n K.

sum(v)

Let u s c a l l A(v)XV a monomial i n A ( X ) .

-

If sum(v) = 1 , then A(v)Xv i s l i n e a r .

2 , then A(v)Xv i s c a l l e d q u a d r a t i c , e t c .

If

Let A ( X ) C 0 , and l e t

its o r d e r , o r d ( A ( X ) ) , b e t h e l e a s t k, in Z ( L 0 ) such t h a t t h e r e e x i s t s a non-zero monomial A(v,)Xvo

i n A ( X ) with sum(v,)

order k, if and only i f A ( X ) =

-

0, and sum(vo) = k,.

mEZ(LO),

+

= m,

and B(X) i n KCCXII,

- --

= k,.

C l e a r l y A ( X ) is of

(Isum(v)=k A(v)x"),

-

with sane A ( v , ) c

Let ord(0) be d e P i n e d t o be -, w i t h > m, for a l l and + n = n + -, f o r a l l nEZ(L0). Given A ( X ) ,

Power series : formal and hyper-convergent

7.40

(1)

(i)

ord(A(X).B(X))

(ii)

ord(A(X) + B ( X ) ) I m i n . ( o r d ( A ( X ) ) , o r d ( B ( X ) ) ) ,

=

ord(A(X))

+

28 1

ord(B(X)),

e q u a l i t y o c c u r r i n g i f o r d ( A ( X ) ) C o r d ( B ( X ) ) , and ord(r)

(iii)

Let M

=

=

0 , f o r a l l reK*.

>

01.

C l e a r l y M i s t h e maximal i d e a l

F u r t h e r , M is t h e i d e a l g e n e r a t e d by X 1 ,

of t h e r i n g K [ [ X ] ] . K[

ord(A(X))

{A(X)EK[[X]]:

... , Xn

in

[ X I 1. Although t h e r e i s no r e a s o n , a - p r i o r i ,

t o t h i n k t h a t we c a n

p v e v a l u a t e f va formal power series A ( X ) , g i v e n as i n (01, a t any o t h e r p o i n t we can d e f i n e

but 0 i n K n

OEK",

evaluated

t o be t h e c o n s t a n t term A ( 0 ) of

'sum(v)=k A(v)Xv).

7.41

NEUMANN SERIES I N A FINITE NUMBER OF VARIABLES

L e t K be any f i e l d , l e t F = K ( ( C ) ) ( r e s p . CK((G))), l e t M be t h e

maximal i d e a l of t h e v a l u a t i o n r i n g 0 of F .

Let nEN, l e t x

=

(xl,

...

, x n ) ~ Mn ,

and l e t S

j

=

s u p p ( x j ) ; then S . is J

a well-ordered s u b s e t of G+ ( r e s p . i s a w e l l - o r d e r e d s u b s e t of C + of power less than

LO

5

1.

L e t S be t h e union of (Sj)lsjsn.

Although a n a b u s e of

n o t a t i o n , s i n c e x is a v e c t o r of elements i n F and hence i s n o t a n e l e m e n t of F , l e t us d e f i n e (0)

s u p p ( x ) t o be S, t h e union of s u p p ( x , ) ,

By Lemma 2 of S e c t i o n 7.21

, we

... , s u p p ( x n ) .

know t h a t S is a w e l l - o r d e r e d s u b s e t

of C+ ( r e s p . i s a w e l l - o r d e r e d s u b s e t of Gt of power l e s s t h a n w 1. 5 n o t e d i n Neumann's Lemma ( 7 . 2 0 ) ,

As w a s

t h e subsemi-group w * S of C g e n e r a t e d by S

282

Norman L . A l l i n g

7.41

is a w e l l - o r d e r e d s u b s e t of Ct ( r e s p . i s a w e l l - o r d e r e d s u b s e t o f

G+

of

power l e s s t h a n w 1.

5

For a l l veZ(LO)", s u p p ( x

(1)

PROOF. vl

t

... + vn V

s u p p ( x ). hold:

Let v

(v

1'

is a s u b s e t of s u m ( v ) - s u p p ( x ) .

... , v n ) ;

t h e n sum(v) h a s been d e f i n e d t o be

Then t h e r e must exist g , ,

... + g n , and

A s we h a v e s e e n (6.201,

V

-

...

.

Let g b e i n

... , g n i n C s u c h t h a t

the following

( 7 . 4 0 ) . By d e f i n i t i o n , x

(1) g = g , +

... , n .

=

V

V

(ii) g

=

j

x1

1

f o r each j = 1 ,

J

s u p p ( x '1

.i

is c o n t a i n e d i n v j * s u p p ( x 1,

J

.i

j

j

'n n

i s i n s u p p ( x ' 1,

whereby O.supp(x ) is meant {O] ( 7 . 2 0 ) ; t h u s e a c h g i n t u r n c o n t a i n e d i n v 0s.

o x

is i n v V S

.lj '

A s a r e s u l t , g is i n v l = S +

...

+

which i s

v n - S , which

is d e f i n e d t o be s u m ( v ) * S (7.20).

we know t h a t , f o r a l l gsC, InsN: g c ( n * S ) l is f i n i t e . A s u s u a l l e t m(g) = 0 , for a l l g s S - w a s ; a n d f o r e a c h g e w * S l e t m ( g ) b e d e f i n e d t o be 1 + max (neN: g s ( n * S ) ] ( 7 . 2 2 ) . Using ( 1 ) we c a n es t a b l i s h t h e f o l l o w i ng By Neumann's Lemma (7.201,

.

(2)

For a l l v ~ Z ( t 0 ) " , w i t h sum(v

PROOF.

Since k

>

m ( g ) , g is n o t i n koS.

s u b s e t of k - S ; t h u s g is n o t i n s u p p ( x v ) .

Let A ( X )

-

a

k -0 ( 'Sun (V ) -k

By ( l ) , s u p p ( x v ) i s a V

Hence, x ( g )

A(v)Xv) be i n

-

0.

KCCXII ( 7 . 4 0 ) .

0

28 3

Power series : f o r m a l and hyper-convergent

7.41

C l e a r l y supp(A(x)

-

A ( 0 ) ) i s a s u b s e t of w.S, w h i c h we know t o b e a

w e l l - o r d e r e d s u b s e t o f G + ( r e s p . a well-ordered s u b s e t of G + power l e s s t h a n w ) ; t h u s A(x) is i n F.

Further, since (2) holds,

5

where we i n t e r p r e t t h e sum of any number of 0 ' s i n ( 4 ) t o be 0.

From t h i s

we see t h a t we have proved t h e f o l l o w i n g .

Let A ( X ) and B ( X ) b e e l e m e n t s of K[[X]]

(7.401, a n d l e t

PEK;

then t h e

following hold:

J u s t a s i n S e c t i o n 7.22, i t i s well t o keep i n mind t h e f a c t t h a t t h e

sum i n ( 3 ) i s always a f i n i t e sum. proved t h e f 011owi ng THEOREM.

.

A(X)EK[CX~, ,

..,Xn]]

Let t h e image of K[[X,

d e n o t e d by K[[x 1

A s a r e s u l t of (5) o n e see t h a t we have

,...,xn]],

+

A(x)EF i s a K - l i n e a r homomorphism.

,...,X n ] ]

( 7 . 4 0 ) u n d e r t h i s homomorphism be

or s i m p l y by K[[x]],

f o r short.

Norman L . A l l i n g

284 7.50

7.50

TRIGONOMETRIC FUNCTIONS

Let K be a f i e l d of c h a r a c t e r i s t i c 0 , l e t F

=

K((G))

( r e s p . CK((G))),

and l e t H be t h e maximal i d e a l of t h e v a l u a t i o n r i n g 0 of F .

Let x be i n

M.

We c a n a l s o d e f i n e g e n e r a l i z a t i o n s of t h e c l a s s i c t r i g o n o m e t r i c f u n c t i o n s , i n t h i s c o n t e x t , as f o l l o w s .

Using Neumann's Lemma ( 7 . 2 0 ) , we see t h a t c o s ( x ) and s i n ( x ) a r e w e l l d e f i n e d elements i n F. (1)

For a l l x and y i n M t h e f o l l o w i n g hold:

(i)

cos(x + y)

=

cos(x)cos(y)

-

sin(x)sin(y),

(ii)

s i n ( x + y)

=

sin(x)cos(y)

+

c o s ( x ) s i n ( y ) , and

(iii) cos2(x)

PROOF.

+

sinz(x)

= 1.

S i n c e s i m i l a r r e s u l t s h o l d f o r t h e c l a s s i c a l s i n e and c o s i n e

f u n c t i o n s over t h e complex numbers, t h e y must h o l d as f o r m a l power s e r i e s i n t w o v a r i a b l e s w i t h r a t i o n a l c o e f f i c i e n t s . Using Theorem 7.41, we see t h a t t h e s e o b s e r v a t i o n s s u f f i c e t o prove ( 1 ) .

o

Note t h a t f o r a l l XEM, (2)

(ii)

c o s ( x ) is i n 1 + H, s i n ( x ) is i n M , and

(iii)

s i n ( x ) = 0 i f and o n l y i f x

(i)

PROOF.

= 0.

( i ) and ( i i ) f o l l o w from (7.34:1), and ( i i i ) f r a n (7.30).

Power series : formal and hyper-convergent

7.50

285

We can d e f i n e o t h e r t r i g o n o m e t r i c f u n c t i o n s a s f o l l o w s :

(3)

tan(x)

=

s i n ( x ) / c o s ( x ) , f o r a l l ; XEH;

cot(x) s ec(x) (iii) (iv) csc(x)

= =

c o s ( x ) / s i n ( x ) , f o r a l l : XEH*; l / c o s ( x ) , f o r a l l XEM;; and

=

l / s i n ( x ) , f o r a l l XEM?.

(i) (ii)

C l e a r l y t h e c o s i n e a n d t h e s e c a n t f u n c t i o n s a r e e v e n , whereas t h e

s i n e , t h e t a n g e n t , t h e c o t a n g e n t , a n d thje c o s e c a n t f u n c t i o n s a r e o d d functions.

C l e a r l y t h e u s u a l addition formula f o r t h e tangent, t h e half

angle formula,

...

, hold f o r t h e s e f u n c t i o n s .

F o r a l l X E M we c a n a l s o

define the following functions:

(5)

(1 -3..

.. (2n -

/ ( 2.4..

.. (2111) ( 2 n + l ) .

( i i)

arcsin(x)

(if

s i n and a r c s i n map M o n t o M, and are i n v e r s e s t o o n e a n o t h e r ,

=

1))

9

( i i ) t a n and a r c t a n map M o n t o M , and a r e i n v e r s e s t o one a n o t h e r .

PROOF. 7.51

The argument u s e d t o prove Theoren 1 , of (7.361, s u f f i c e s . ELEMENTARY FUNCTIONS OVER REAL A N D COMPLEX CONSTANT FIELDS

Assume now t h a t K gC((C))).

0

- R.

Let u s i d e n t i f y F ( i ) w i t h C ( ( C ) ) ( r e s p .

Let W be t h e e x t e n s i o n t o F ( i ) of t h e v a l u a t i o n V of F , d e f i n e d

i n (7.1 1 :6). Consider t h e f o l l o w i n g c l a s s i c a l e n t i r e f u n c t i o n s : t h e e x p o n e n t i a l f u n c t i o n , zcC

Z

e cC*, t h e c o s i n e f u n c t i o n , and s i n e f u n c t i o n . Let O c x be t h e v a l u a t i o n r i n g of W a n d l e t Flex be its m a x i m a l i d e a l +

( d e s c r i b e d i n a n o t h e r way i n ( 7 . 1 1 : 8 ) ) . (0)

(1)

C l e a r l y we have t h e f o l l o w i n g :

For wgOCx t h e r e e x i s t unique CEC and Zencx s u c h t h a t w

( i i ) For ucO t h e r e e xist unique rcR and x€Hcx such t h a t u

-

=

c

+

r

+

x.

z.

7.51

Norman L. Alling

286

Let us extend the exponential function from C to using (O,i), let Exp(w)

Exp(c

=

+ z)

=

ocx as

follows:

ec-exp(z) (7.36), for a l l weOcx.

Then, using classical results, and those of Section 7.36, one can see that (1)

(i)

Exp maps Ocx onto C**(l

(ii)

for all w, and w 1 in Ocx, Exp(w,

(iii)

~ x pis a one-to-one mapping of

(iv)

for a l l W E O ~ and ~ , for all neZ, Exp(w + 2nin)

(v)

Exp(w)

1

=

+

Mcx);

if and only if w

=

w,)

+

o

=

Exp(w,).Exp(w,);

onto R + - ( I + MI: =

Exp(w); and

2nin, for some ncZ.

Given WEO let w = c + z (O,i), and define an extension of the cx’ cosine and sine as follows: let (2)

(i)

(ii)

= =

Cos(c Sin(c

+ z) = + z) =

cos(c)cos(z) sin(c)cos(z)

- sin(c)sin(z), and let +

cos(c)sin(z).

extended t o Ocx, these functions have the following properties:

As

(3)

Cos(w) sin(w)

Cos(w,

(ii)

Sin(w, + w,) = Sin(w,)Cos(w,) + Cos(w,)Sin(w,); and Cos2(w) + Sin2(w) = 1, for all w o , w l , and w in Ocx.

(iii)

PROOF.

w,)

+

=

~os(w,)Cos(w,)

- Sin(w,)Sin(w,);

(i)

COS(W, +

(i),

W,) = cOS(C, + C, + Zo + 2 , )

-

cos(c,

+

c,)cos(z,

+ z,)

- sin(c, -

( c o s ~ c , ~ c o s ~-c sin(c,)sin(c, ,~ ))(cos(z, (sin(c,)cos(c,)

+

+

c,)sin(z,

+ z,)

)cos(z,) - sin(z,)sin(z.,

cos(c,)sin(c,))(sin(z,)cos(z,)

+

1)

-

cos(z,)sin(z,))

cos(c,)cos(c,)cos~z,~co~~z,) - c o s ~ c , ~ c o s ~ c , ~ s i n ~ z , ~ s-i n ~ z , ~

sin(c, )sin(c, )cos(z, )cos(z,)

+

sin(c, )sin(c, )sin(z, )sin(z, 1 -

- sin~c,~cos~c,)cos~z,)sin~z, 1cos(c,)sin(c, )sin(z,)cos(z,) - cos(c,)sin(c, )cos(z,)sin(z, 1

sin(c,)cos(c, )sin(z,)cos(z,)

-

Power series : formal and hyper-convergent

7.51

)COS(Z,

COS(C,)COS(Z,)COS(C,

287

1 - cos(c,)cosfz,)sin(c,)sin(z,)

sin(c, )sin(z, )cos(c, )cos(z,

+

sin(c,)sin(z, )sin(c, )sin(z,)

- sin(c,)cos(z,)cos(c,)sin(z,) )sin(c, )cos(z, 1 - cos(c,)sin(z,)cos(c, )sin(z, ) -

sin(c,)cos(z,)sin(c,)cos(z,) cos(c, )sin(z,

( c o s ~ ~ , ~ c o s- ~s zi n, (~c , ) s i n ( z , ) ) ( c o s ( c , ) c o s ( z , ) (sin(c,)cos(z,)

+

-

-

sin(c,)sin(z,))

-

cos(c,)sin(z,))(sin(c, )cos(z,) + cos(c,)sin(z,))

cos(c,

+

z,)cos(c,

+

z l ) - sin(c,

+

z,)sin(c,

+

z,)

-

Cos(w,)Cos(w,)

-

Sin(w,)Sin(w,);

establishing ( i ) . For a more conceptual p r o o f , n o t e t h a t ( i ) c o u l d be deduced f r o m t h e f a c t t h a t t h e a d d i t i o n formulas f o r t h e c o s i n e and t h e s i n e f u n c t i o n s o v e r C are e q u i v a l e n t t o similar s t a t e m e n t s a b o u t formal power s e r i e s w i t h

rational coefficients, i n several variables.

These s t a t e m e n t s , a f t e r

s u i t a b l e s u b s t i t u t i o n s and a p p e a l t o r e s u l t s proved i n t h i s C h a p t e r , i m p l y ( i ) . S i m i l a r p r o o f s may be g i v e n f o r ( i i ) and ( i i i ) .

(4)

For a l l zcMCX, E x p ( i z ) = Cos(z) + i S i n ( z ) .

R e c a l l i n g t h e f a c t t h a t (7.22:2)

PROOF.

Exp(iz) =

m

n ( i z ) /n! =

(-1)"(2)~"/(2n)! +

(5)

lnIo( i z l 2 " / ( 2 n ) !

&Io ( i ~ ) * ~ + l / ( 2 n + l ) ! =

i*lnIo( - 1 ) ~ ( 2 ) ~ ~ + ~ / ( 2 n =+ lc )o !s ( z )

For all W E O ~ ~ Exp(iw) ,

PROOF.

+

i s a f i n i t e sum, we s e e t h a t

-

+

iSin(z).

Cos(w) + i S i n ( w ) .

Exp(iw) = E x p ( i c + i z )

-

eiC*Exp(iz)

I

(cos(ic) + isin(ic))*(cos(iz)+ isin(iz)) I

( c o s ( ic ) c o s ( iz)

- s i n (ic ) s i n ( iz))

+ i ( s i n ( ic)cos ( iz ) + cos (ic ) s i n ( iz 1) I

cos(iw)

+

iSin(iw).

o

o

288 7.60

DERIVATIVES OF FORMAL POWER SERIES

L e t K be any f i e l d , l e t F = K ( ( G ) )

( r e s p . gK((G))),

. ..

n ,xn)€M

.

and l e t M be t h e

Let n be i n N and l e t x =

maximal i d e a l of t h e v a l u a t i o n r i n g 0 of F.

(x,,

7.60

Norman L. A l l i n g

and l e t A(x) be i n K[[x]].

Let A ( X ) be i n K[[X]],

L e t us

S e c t i o n s 7.40 and 7.41 f o r n o t a t i o n a l conventions and d e f i n i t i o n s . ) d e f i n e t h e formal p a r t i a l d e r i v a t i v e , a A ( X ) / a X i ,

of A ( X ) t o be

V

IkmO (Isum(v)=k v 1. A ( v l ,

(See

... ,vn)X1 1 ... -Xi

v

i

- 1 *

...

V

").

o x n

C l e a r l y a l l t h e f a m i l i a r p r o p e r t i e s of p a r t i a l d e r i v a t i v e s h o l d f o r formal p a r t i a l d e r i v a t i v e s : e . g . , partial derivatives,

... .

K - l i n e a r i t y , c o m r n u t a t i v e l y o f mixed

F u r t h e r , T a y l o r s e r i e s expansions of formal

power series e x i s t and have t h e f a m i l i a r p r o p e r t i e s .

Let u s c o n s i d e r t h e case i n which n l e t dA(X)/dX = k- 1 Ikml ka)(x =

.

(A(k)(x))t,

lkI, kakXk- 1 .

-

Let A(X)

1.

Assume now t h a t XEH.

f o r a l l kcZ(2O), A(k)(0)

=

Note t h a t A(0)

=

lkm akXk, O

and

Let dA(x)/dx =

Further, l e t

T h i s w i l l a l s o be denoted by A ' ( x ) .

f o r all keZ(L0).

=

a o , A'(0)

=

A

(k+l)

a l , and t h a t

k!ak; t h u s we have t h e f o l l o w i n g :

lkIo (A(k)(0)/k!)Xk,

and A(x) =

lkZO

( A ( k ) ( 0 ) / k ! ) xk

.

(2)

A(X)

(3)

D i f f e r e n t i a t i o n commutes w i t h t h e K - l i n e a r s u b s t i t u t i o n homomorphism X

=

=

(X,,

.. . , X n )

+

x

=

(x,,

.. .

, xn) (7.41).

Power series: formal and hyper-convergent

7.61 7.61

28 9

INFINITESIMAL EXTENSIONS OF A N A L Y T I C FUNCTIONS, I

L e t F = C((C)) ( r e s p . gC((G))) and l e t U be a non-empty open s u b s e t Let f be an a n a l y t i c f u n c t i o n on U.

of t h e complex p l a n e C .

For e a c h C E U ,

f c a n , of c o u r s e , be w r i t t e n as f o l l o w s :

-

f o r a l l ~ E Ca n d 1s l e t z be i n Mcx.

cI




-

x

+

x

-

x , ) L min.(V(x,

-

t h e n h = V(x,

x ) , V(x

-

-

x,)

=

x l ) } L min.{g,g}

g ; which i s

=

o

absurd.

Addition i n F is c o n t i n u o u s , i n t h e m o d i f i e d v a l u a t i o n

LEMMA 1 . topology

. Let x, a n d y , € F ,

PROOF.

Then V ( ( x

y~B(y,,Lg). min.{V(x

=

S u p p o s e , f o r a mo me n t , t h a t t h e r e i s a

h.

p o i n t x i n B(x,,Lg) a nd i n B ( x,,Lg) ; V(x,

- x,)

a n d x , be d i s t i n c t p o i n t s i n F , a n d l e t V(x,

Let gEG s u c h t h a t g

-

x,),

V(y

-

+

Y)

a n d l e t geC.

- (x,

+

Let x e B ( x , , L g )

y o ) ) = V((x

y o ) ) ] t g; showing t h a t x

+

-

x,)

YEB(X,

+

(Y

+

and let

-

yo)) L

y,,Lg).

M u l t i p l i c a t i o n i n F is continuous, i n t h e modified valua-

LEMMA 2 .

t i o n t o p o l o g y on F.

Let x, and y,cF,

PROOF.

l e t gEC, a n d l e t hEC s u c h t h a t h

(2)

Let YEF be s u c h t h a t V(y

-

yo) L g

(3)

L e t XEF be s u c h t h a t V(x

-

x,) L max.{g

-

xy,

Then V(xy v((x

-

x,)y,)}

-

x,y,)

= V(xy

= min.IV(x)

+

V(y

-

+

-

>

V (x , ).

V(x,).

xy,

y o ) , V(x

-

-

-

V(Y,), h l .

x,~,)

L min.{v(x(y

x,)

V(y,)l.

+

Lemma 2 of S e c t i o n 7.62, ( 2 ) a n d (3) a g a i n , we see t h a t V ( x y

-

Y,)),

Applying

-

(31,

x o y o ) 2 g.

0

LEMMA

3.

t o p o l o g y on F.

Division i n F is continuous i n t h e m o d i f i e d valuation

Norman L . A l l i n g

294

Let x,EF*, and l e t gEG, and l e t hEG s u c h t h a t h

PROOF.

Let XEF* s u c h t h a t V(x

(4)

7.64

-

Then V ( l / x

l/x,,)

=

-

x , ) L min.{g

V((x, - x)/xx,)

Lemma 2 of S e c t i o n ( 7 . 6 2 1 , V( / x

( 4 ) , we see t h a t V ( l / x

-

-

l / x o ) L g.

l/x,

V(x,).

2V(x,), h ) .

+

=

>

V(x, - X ) - V(X) - V ( X , ) .

-

V(x,

-

-

x)

2V(x,).

By

Applying

0

Combining t h e s e r e s u l t s we see t h a t we have proved t h e f o l l o w i n g .

F is a t o p o l o g i c a l f i e l d , i n t h e modified v a l u a t i o n

THEOREM 0 .

topology on F .

Let F be a n o r d e r e d f i e l d , l e t V be t h e o r d e r - v a l u a t i o n on F (6.00) and l e t G be t h e v a l u e group of V. THEOREM 1 .

The i n t e r v a l t o p o l o g y o n F a n d t h e m o d i f i e d v a l u a t i o n

topology on F a r e i d e n t i c a l .

PROOF.

Let gEG.

S i n c e B(0,Lg) (7.62) i s an o p e n i n t e r v a l i n F , we

s e e t h a t i t i s a n open s e t i n t h e i n t e r v a l t o p o l o g y .

Let I be a non-empty

S i n. c e I i s an open i n t e r v a l i n F , t h e r e e x i s t i n t e r v a l i n F , and l e t ~ ~ € 1

x , a n d x , i n I , f o r w h i c h x, < x1 V(x,

- x,).

Let h

>

max.{g,,

< x,.

Define g o

PROOF.

-

x , ) and g, =

L e t X,EF and l e t gEG.

B(x,,Lg)

Clearly t h e u n i o n of ( ( x o

-

Hence I

o

i s a c-open s u b s e t of F.

nu-g, x,

+

where u - ~i s d e f i n e d t o be an element i n F such t h a t g.

V(x,

g 2 } ; t h e n B(x,,Lh) is a s u b s e t of I .

is a n open s e t i n t h e i n t e r v a l t o p o l o g y o n F. LEMMA 4.

=

nu -g ) ) n E N is Bfx,,Lg), U J - ~

>

0 and V ( U J - ~ ) =

0

Let u s d e f i n e t h e c-topology g e n e r a t e d by [ B ( x , , > g ) : gEG, X,EFI t o be t h e m o d i f i e d c-topology o n F .

Each s e t i n t h i s s e t of s e t s w i l l be c a l l e d

a modified C ws u b s e t of F.

As a consequence of Lemma 4, we see t h a t we

have proved t h e f o l l o w i n g .

7.64

Power series : formal and hyper-convergent

29 5

Each s e t i n t h e m o d i f i e d c - t o p o l o g y o n F i s i n t h e 5-

THEOREM 2.

t o p o l o g y on F.

(5)

For grC there i s no l e a s t element YEF s u c h t h a t B(0,Bg)

(i)

( i i ) For grG t h e r e i s no g r e a t e s t element

( i i i ) For a

< [y].

z i n B(0,Lg).

< bEF, no x o c F and no gcC e x i s t f o r ( a , b )

=

B ( x o , h g ) ; and

t h e r e is no X,EF and no gEG s u c h t h a t [ a , b ] = B ( x , , L g ) .

(iv)

S i n c e B(0,Lg) is a non-zero convex s u b g r o u p o f ( F , + ) , t h e r e

PROOF.

i s no l e a s t element ycF s u c h t h a t B(O,2g)

< { y ] , and no g r e a t e s t e l e m e n t

z

i n B(0,Lg); p r o v i n g ( i ) and ( i i ) . Concerning ( i i i ) , s u p p o s e f o r a moment t h a t s u c h x,

and g e x i s t ; t h e n (a

-

xo,b

-

x,) = B(O,Lg), which v i o l a t e s

( i ) . Concerning ( i v ) , s u p p o s e f o r a moment t h a t s u c h xo and g e x i s t ; t h e n

[a

-

x,,b

-

x,]

=

B(O,hg), which v i o l a t e s ( i i ) .

Let a

EXAMPLE 1 .


O)"-l,

there are only a

7.71

Power series: formal and hyper-convergent

29 9

f i n i t e number of terms i n ( * ) of degree u ( 7 . 4 0 ) ( c f . ( 7 . 7 0 ) . quence we s e e t h a t ( * ) i s a wel.1-defined formal power series.

As a

conse-

( C f . Section

7.35.) PROOF.

Note t h a t A(0,

-F(X)/A(O,

... , 0,

(l)

=

1).

... , 0,

1 ) = ( a F / a X n ) ( 0 ) f 0.

Let C ( X ) =

Then, V

- 'n

+

1 ... .x n ... , (0, . . , 0, 1 ) .

lkml l s u m ( i ) = k B(vl,

where t h e prime i n d i c a t e s t h a t v A

W e m u s t show t h a t t h e r e e x i s t s a unique g ( X l , K[[X,,

... , X n _ , ] ] ,

t

G l Llln(v)=k

B(vl,

... , vn)X1

'

... , X n e l )

in

V .. 'xn-l n- 1 *(g(X , ... , X n - 1 ) )

V

1.

0,

Let us examine t h e l i n e a r terms of g.

Since v A (0,

n

such t h a t t h e f o l l o w i n g holds:

where t h e primes i n d i c a t e t h a t v b

k.

V

Vn)X1

... , 0,

... , 0,

'n

,

1)

These o c c u r o n l y f o r m = 1

=

1 ) i n ( 2 1 , t h e l i n e a r terms i n g do n o t i n v o l v e

any of t h e c o e f f i c i e n t s of g i n t h e s e c o n d e x p r e s s i o n i n ( 2 ) ; h e n c e i f sum(u) = 1 , C(u) i s a polynomial i n t h e B ( v ) s , w i t h c o e f f i c i e n t s i n 2.

Having d e a l t w i t h t h e l i n e a r terms i n ( 2 ) above, l e t U O Z ( L O ) ~ - ~ .w i t h sum(u) = m

>

1 ; t h e n C ( u ) X u i s a n o n - l i n e a r term i n t h e f i r s t e x p r e s s i o n i n

(2), which m u s t e q u a l t h e sum of

second expression i n (2). k , and assume t h a t v b ( 0 ,

terms, g i v e n by v a r i o u s V O Z ( L O ) " i n t h e

Let VEZ(LO)" be s u c h a n element.

... , 0,

1).

Let sum(v)

I f , f o r a l l s u c h v , vn

-

=

0, then

t h e r e a r e n o c o e f f i c i e n t s C(u), w i t h sum(u) = m, i n t h e second e x p r e s s i o n in ( 2 ) . Thus, C(u) is a polynomial i n t h e B ( v ) s , w i t h c o e f f i c i e n t s i n Z.

7.71

Norman L . A l l i n g

300

Our c o n c e r n is w i t h

Assume now t h a t s u c h v e x i s t f o r which vn b 0 .

Let psZ(LO)"-'

t h e c o e f f i c i e n t s , C(ul) of s u c h terms.

... , n

for a l l j

=

qcZ(L)"-l

and u

1,

=

p

-

Then p 5

1.

u in Z ( L 0 )

n- 1

.

such t h a t p j

Let q = u

-

=

v

j'

p; t h u s ,

Then we see t h a t t h e f o l l o w i n g i s a term i n Xu i n

+ q.

t h e second e x p r e s s i o n i n ( 2 ) :

(3)

... , vn)X1P1 ... ... , qhEZ(20)n- 1 , surn(qj

B(vl, ql,

for all j and w i t h p . = v J j' Since p definition

(4)

Pn- 1C ( q l ) *

axn-

+

q

lj=lq j

sum(qi)




... , 0,

F u r t h e r , s i n c e , by

0 ; hence p f 0 , and t h u s s u m ( p )

sum(u)

-

sum(p)

t h a t vn

>

1.

sum(qi)




0.

... , n =

Since each sum(q.) 2 1 and s i n c e v n ( = h ) J sum(qh)

=

0.

Assume

- 11. such

Clearly sum(qi) 5 s u m ( q ) =

sum(u); e s t a b l i s h i n g ( 4 1 , i n case v n

+

>

1 ) ( ( 1 ) a n d (211, and s i n c e , by

1 , t h e r e must e x i s t a j c { l ,

>

...

vn'

... , h.

that v. J




L a s t l y assume 1 , we s e e t h a t

sum(q) 5 sum(u); e s t a b l i s h i n g ( 4 ) .

Using ( 4 ) and t h e e a r l i e r r e s u l t s o b t a i n e d i n t h i s S e c t i o n , we s e e that

(5)

C(u) i s a polynomial w i t h c o e f f i c i e n t s i n Z i n t h e

B(V)'S

C(ul)s, f o r which sum(ul) < sum(u). T h u s , by i n d u c t i o n on s u m ( u ) , t h e Theorem has been proved.

and t h e

Power series: formal and hyper-convergent

7.71

301

The development i n t h i s S e c t i o n i s q u i t e c l o s e

BIBLIOGRAPHIC NOTE,

t o o n e g i v e n b y Gunning a n d R o s s i [40, p p . 14-151, s t r i p p e d of c o u r s e of all analysis.

7.72

THE FORMAL IMPLICIT MAPPING LEMMA

Let K be a f i e l d .

Let k and n be i n N , w i t h k

... , f n E K [ [ X 1 , ... , X n ] l a l l j = k + 1 , ... , n; and

Let f k + l ,

LEMMA.

for

n.

K[[XIl,

(i)

f.(O) J

(ii)

( a f . / a X i ) ( 0 ) = 6 : . f o r a l l i and j = k + 1 , , n. J , Xk) i n K[[Xl, Then t h e r e e x i s t unique g ( X

such t h a t

...

all

j = k

( i i i ) g.(O)

J

(iv)

= 0,

=




0:

Norman L. A l l i n g

322

k -n In=, w /n!

(I )

+ w

-w

+

Thus, V(ea - Sk) Further, e

(Sk)osk.

., .

+

terms of larger v a l u e .

k + 1 , f o r a l l ksN.

a

7.90

Hence ea i s a pseudo-limit of

is t h e simplest pseudo-limit

of (Sk)osk,

t t s i m p l e s t t *i s u s e d in t h e s e n s e of Conway [24, p. 231.

I,,,m

(w

k (In,,

m

7.91

(See a l s o ( 6 . 4 1 )

We conclude t h a t

and (6.431.)

(2)

where

-1

(w

w

-n

+

w

-1

-W

w

+

-W

n

In! is

not

t h e s i m p l e s t pseudo-limit of

n 1

/n! is.

FROM MACLAURIN SERIES TO TAYLOR SERIES

Let K be a f i e l d of c h a r a c t e r i s t i c 0 , a n d l e t G be a n o n - t r i v i a l

ordered group.

Let F

K ( ( G ) ) ( r e s p . gK((G)).

=

i d e a l i n t h e v a l u a t i o n r i n g 0 of F.

in K , and l e t y be i n M.

In:o

(0)

L e t M d e n o t e t h e maximal

L e t (an)Osn be a sequence of e l e m e n t s

By Neumann's Theorem,

anyn is a w e l l - d e f i n e d element i n F.

Let ( 0 ) b e d e f i n e d t o be a Maclaurin-Neumann series.

i n F such t h a t x

f(x)

f1)

-

- lnlO OD

an(x

-

x,)" is a w e l l - d e f i n e d element i n F,

which we w i l l d e f i n e t o be a Taylor-Neumann series. x1

-

x o is i n M.

Let x and x, be

x, is i n M ; t h e n

Let X ~ E sFu c h t h a t

Consider t h e f o l l o w i n g well-defined element i n F:

Power s e r i e s : formal and hyper-convergent

7.91

(3)

Let bk

In=, ( n+k )*an+,(x, m

=

x,)

n

32 3

EF.

We would l i k e t o a r g u e t h a t t h e l a s t e x p r e s s i o n i n ( 2 ) e q u a l s t h e

following:

w h i c h we would l i k e t o d e f i n e ; however, since we do n o t know t h a t t h e b k f s

a r e i n K , we can n o t invoke Neumann's Theorem t o e v a l u a t e ( 4 ) ! T h e c o n t e x t t h a t i n t e r e s t s u s t h e m o s t , of c o u r s e , is t h e o n e i n which t h e power s e r i e s f i e l d F is CNo o r ~ C X . I n t h e n e x t S e c t i o n we w i l l

c o n s i d e r t h i s q u e s t i o n s over t h e f i l e d L , d e f i n e d i n S e c t i o n 7.82.

FROM MACLAURIN SERIES TO TAYLOR SERIES OVER L , I

7.92

L e t t h e s e t t i n g be as i t was i n S e c t i o n 7.82, w i t h t h e e x c e p t i o n t h a t

we w i l l assume i n a d d i t i o n t h a t t h e ground f i e l d K has c h a r a c t e r i s t i c 0 . Let ( a n ) 0 6 n be a s e q u e n c e of e l e m e n t s i n L ( 7 . 8 2 : 0 ) ,

and consider the

f o l l o w i n g formal power series:

In10 any

i n L"YII.

Let P be t h e prime d i s k of hyper-convergence of ( 0 ) ( 7 . 8 4 ) ; t h e n

f o r a l l PEP, f , ( p )

L e t P E P , x ~ E L ,and l e t x

f(x

=

an(x

n

=

- x,) n

anp =

p

+

i s a w e l l - d e f i n e d element i n L .

x,.

Note t h a t x

- xg

=

is a well-defined element i n L .

PEP.

Hence,

324

Norman L. A l l i n g

7.92

P + x, w i l l be d e f i n e d t o be t h e prime d i s k of h y p e r - c o n v e r g e n c e o f

f ( x ) (cf. (7.84)).

R e c a l l t h a t g i v e n any p r i m e i d e a l PI of 0 , t h a t i s

c o n t a i n e d i n P , t h e n PI

+

x, i s c a l l e d a prime d i s k of hyper-convergence of

f ( x ) (7.84). L e t f ( k ) ( x ) be t h e k ' t h f o r m a l d e r i v a t i v e of f ( x ) ; w h i c h i s t h e

following:

(3)

m

ln,kn(n - l ) ( n - 2).

Note P

+

...*( n

-

k + l ) a (x

n

- x,)"-~.

x, i s a l s o t h e p r i m e d i s k o f h y p e r - c o n v e r g e n c e of f ( k ) ,

s i n c e t h e v a l u e of t h e elements i n Z * i s always z e r o . t h a t x1

(4)

-

x,EP.

(i)

bk =

Let X ~ E Lbe s u c h

By t h e Main Theorem (7.821, we know t h a t

-

lnPo(n+k ) * a n + k ( x l -

( i i ) Note a l s o t h a t bk

-

x,)" i s a w e l l - d e f i n e d element i n L.

f ( k ) ( x l ) / k ! , f o r each ksZ(20).

The e x p r e s s i o n s on t h e r i g h t i n ( 5 ) a r e power s e r i e s e x p a n s i o n s i n L:

We want t o c o n s i d e r t h e f o l l o w i n g :

Recall t h a t i t was e x a c t l y a t t h i s p o i n t t h a t we reached a n impasse i n S e c t i o n 7.91.

F u r t h e r , r e c a l l t h a t i n S e c t i o n 7.82 we d e f i n e d A t o be

t h e c a n o n i c a l d i r e c t summand o f B i n G, and noted t h a t t h e a n ' s (0) a r e a l l

in EK((B)).

Note a l s o t h a t t h e power s e r i e s i n ( 1 1 ,

c o n s i d e r e d t o be i n E K ( ( B ) ) ( ( A ) ) .

(21, a n d ( 3 ) may be

The problem t h a t c o n f r o n t s u s in (6) is

7.92

325

Power series: f o r m a l and hyper-convergent

t h a t t h e c o e f f i c i e n t s b k n e e d n o t b e i n CK((B))!

In order avoid t h i s

d i f f i c u l t y , l e t u s proceed a s f o l l o w s . Let 8

v(i)

< w 5'

=

[ h ( a , i ) : i = 0, 1 , and a

f o r i = 0 and 1 ; t h u s 101

subgroup of G t h a t contains

r

<


m(z),

h , which

n+k )*an+,(x, then { (

is j u s t sum(k,n).

-

x,)

n

*(x

-

x,)

k

}(z) =

Thus, we have t h e f o l l o w i n g :

This being t h e c a s e one s e e s t h a t t h e f o l l o w i n g is t r u e : D ( y ) ( z ) =

IkIo

(Inso{(n+k k )'an+k(xl

g ( x ) ( z ) , since supp(x

-

x,)

xO)nm(x

( = S,)

-

k '1)

I('))

=

lk,O (la,(' -

is a s u b s e t of S .

Thus,

xl)k}(z)

=

Power series: f o r m a l and hyper-convergent

7.92

S i n c e x - x, S, f ( x ) ( z ) = =

0.

lj:o

=

x - x1

+

327

x i - x , , and s i n c e S , a n d S, a r e s u b s e t s of

{ a . ( x - x a ) J ] ( z ) ; and f o r a l l j J

>

m(g),

[a.(x J

-

x,)j](z)

Thus, t h e f o l l o w i n g sums have o n l y a f i n i t e number of non-zero terms:

f(x)(z)

=

ljlo l a j ( x

- x,) j ~ ( z )=

ljIo Ia:C(x

- x,)

+

(x,

-

x , ) l j ~ ( z )=

J

lj=o

{aj.lkJ=o(:I(.

lkIo (In=, I( m

- x,) k ( x , - x , ) j - k

n+k ).an+,(x,

- x,) n - ( x

-

I(Z)

=

x , ) k } ( z ) ) = D(Y)(z); t h u s

Taken t o g e t h e r , ( 1 5 ) and ( 1 6 ) p r o v e t h e Theorem. 7.93

o

FROM MACLAURIN S E R I E S TO TAYLOR SERIES OVER L , I1

Let t h e s e t t i n g b e as i t w a s i n S e c t i o n 7,82,

with t h e exception that

we w i l l a g a i n assume t h a t t h e ground f i e l d K has c h a r a c t e r i s t i c 0.

In this

S e c t i o n we w i l l g e n e r a l i z e t h e r e s u l t s o b t a i n e d i n t h e last s e c t i o n t o Taylor-Neumann series i n s e v e r a l v a r i a b l e s . S i n c e t h e proofs are v i r t u a l l y t h e same a s t h o s e g i v e n i n S e c t i o n 7 . 9 2 ,

t h e y w i l l h e r e be s l i g h t l y

abbreviated.

L e t A(v) have t h e f o l l o w i n g power series e x p a n s i o n i n L:

Let B be t h e smallest convex s u b g r o u p o f G t h a t c o n t a i n s t h e s e t { g ( a , v ) : v ~ Z ( t 0 ) " and a


B}.

we saw i n S e c t i o n 7.82,

Irl
B!']

o f t h e tower of f i e l d s , (C#No: 611

=

{V(x)}

Let B!' b e t h e analogue o f B'

(7.92:2).

Since No is t h e union

a + 1 , f o r a ~ O n ) ,t h e r e is a p o s i t i v e

regular index gS, f o r which Cli L 5, such t h a t x,~c#No. Thus x, and x1 a r e

i n CtNo and f ( X ) i s i n cINo[[X]]. Let Pi/ be t h e i n t e r s e c t i o n of P! w i t h SIINo. Now l e t P#' be defined f o r .$No, as P I was d e f i n e d i n ( 7 . 9 2 : 7 ) f o r CNo.

Let XEPI' + x , .

By Theorem 7.92, t h e following holds:

T h u s f i s a n a l y t i c a t x, i n No, i t s i n t e r v a l of hyper-convergence about x ,

being P!'

+

x1 .

0

336

Norman L . A l l i n g

8 -01

LOCAL PROPERTIES OF POWER SERIES I N ONE V A R I A B L E , I

8.01

Assume f i r s t t h a t (0)

F is a f i e l d w i t h v a l u a t i o n V^, having v a l u e g r o u p A .

Let x , , x,

- xo

x , , and x, be i n F , with x, f x,.

Let V^(x, - x , )

c a n n o t b o t h be z e r o .

By ( * ) n o t b o t h a , and a, c a n be

(1)

For a l l n L 2 , (x, n- 1 ((xz - xo) + (x,

-

x,)

-

(x,

n-2.

(1 +

... +

z+

t h e n un

-

(un-l

Un-2

+

z

vn = - ( v v

(2)

(i)

v)(un-l

n

-

-

-

x,)

F u r t h e r , if a ,

-

v^((x,

PROOF.

-

V^(x,

-

-

... +

z)(l

-

vn

-

... + v

zn )

n-1

n-1 + u " - ~ v + u)*(u

n- 1

... +

z +

+

un(l

- x,)

(x,

+

n- 1

Assume,

+

x,)"

(x,

min.{(n

lishing (2,i). (6.00:3,ii).

=

a,.

n u (1

=

1.

).

n- 1 z ).

-

Let

2)-

L e t ueF and VEF*;

... + v n - l )

=

-

V).

x,,,

L v^(x,

-

x,)

+

(n-1

-

-

x,)

+

(n-1 * m i n . ( a l , a z l .

and l e t v = x ,

-

(u

-

x,

=

+ a,,

-

(x,

-

x,)") then

-

xoln)

v^(x,

Using ( 1 ) a n d t h e t r i a n g l e i n e q u a l i t y ( 6 . 0 0 : 3 , 1 ) ,

x,)

x,)

-

un-*v +

Now l e t u

(x,

t h e f o l l o w i n g h o l d s : V^((x, V^(((X,

x,)

x,.

o

we see t h a t :

V"((X,

(it)

+

-

u ) = -(v

... + vn-1 ).

+

From ( 1

n

-

(u

) =

-

and V^(x,

- x,).

= (x,

x,) +

= (1

ucF* a n d VEF, and l e t z = v / u ; t h e n un n

x,)"

-

(x,

zn

-

For any z i n F, 1

PROOF.

- a,

x g and

m.

-

x,)"

-

Note ( * ) t h a t x 1

-

x,)

-

x,)

n-2,

- l)*a,,

n

-

(x, (n

-

(x,

-

x p )

-

x,)") +

...

2)*a, + a , ,

= +

vA(xz

(x,

... ,

-

-

x,) +

xo)n-l)

(n

-

we see t h a t

2

l ) - a , } ; estab-

To e s t a b l i s h ( i i ) , u s e ( 1 ) a n d t h e t r i a n g l e e q u a l i t y

n

in addition, t h a t

(3)

3 37

A primer on a n a l y t i c f u n c t i o n s o f a s u r r e a l v a r i a b l e

8.01

k is a f i e l d , 6 is a p o s i t i v e r e g u l a r i n d e x , F = g k ( ( A ) ) , and t h a t V-

i s t h e Hahn v a l u a t i o n o f F , having v a l u e g r o u p A; w i t h v a l u a t i o n r i n g 0-, and maximal i d e a l Me.

Let f,(X)

=

In:,

By Neumann's Theorem ( 7 . 2 2 1 ,

c .Xn be i n k[[X]].

n

f , ( x ) is a w e l l - d e f i n e d e l e m e n t o f F, f o r a l l XEM-. Let x

=

m

+

x,EM^

x,,.

+

Let c, a n d x, be i n F.

lnmoc;(x

Let f ( x ) be d e f i n e d t o be

-

x,)

n

,

f,(x - x,). N o t e f u r t h e r t h a t f ( x ) i s a welld e f i n e d e l e m e n t o f F ; and t h a t f is a map o f M e + x, i n t o F. By ( 7 . 3 4 : l )

which is e q u a l t o c ,

+

we know t h a t f maps Me + x, i n t o Ma

c,.

Assume t h a t e l # 0; t h e n f is a n i n j e c t i o n .

LEMMA 0. PROOF.

+

Assume t h a t x , a n d x, a r e i n M A

+

x,,

w i t h x , f x,.

By

( 7 . 3 1 : l ) a n d ( 1 1 , we have t h e f o l l o w i n g .

PROOF.

=

supp(x,

-

Thus, t h e support

x,). Of

S i n c e x , , x,EM-

In:2

C n * ( I jn-o 1

i n w * S (7.411, and h e n c e ( f ( x , ) lishing (4). LEMMA 1 .

=

supp(y).

S is t h e u n i o n of S, a n d S z , where S,

d e f i n i t i o n (7.41:0), a n d S,

- x o , x, - x,), a n d l e t S

Let y = (x,

-

(X,

+ x,,

-

f(x,))/(x,

S i n c e c , # 0, we see t h a t f ( x , )

supp(x,

-

x,),

S, a n d S, are s u b s e t s o f A'.

-

x,)')

is c o n t a i n e d

- x , ) i s i n MA - f ( x , ) 4 0. o

+ c , ; estab-

n-1 -j. ( x l

X,)

=

Then, by

Assume f u r t h e r t h a t k is a n o r d e r e d f i e l d .

Let F be g i v e n

t h e l e x i c o g r a p h i c o r d e r i n d u c e d o n i t by t h e o r d e r o n k a n d o n its v a l u e

g r o u p A. (i)

If 0

( i i ) If 0

Assume t h a t c , # 0, and t h a t x ,


c l , then f ( x , ) > f ( x , ) .

PROOF.

S i n c e t h e o r d e r o n F is t h e l e x i c o g r a p h i c o r d e r , a n d s i n c e

( 4 ) h o l d , we see t h a t f ( x , )

-

f ( x , ) is p o s i t i v e o r n e g a t i v e a c c o r d i n g as

33 8

8.01

Norman L . A l l i n g

-

c,(x,

x , ) is p o s i t i v e o r n e g a t i v e .

Assume t h a t c ,

LEMMA 2.

0.

~ , ( Y ) E F " Y I I such t h a t f o r y

There e x i s t s a unique

(i)

+

g ( y ) , d e f i n e d t o be g , ( q )

+

x,,

= q +

c,EM"

+ cot

i s a w e l l - d e f i n e d e l e m e n t i n MA + x,,

f o r which f ( g ( y ) ) = y. (ii)

x, f o r a l l XEM" + x,. maps M" + x, onto MA + c o , and g maps M A

g(f(x))

=

( i i i ) Thus, f

PROOF.

-

Let H(X1,X2) =

H(0,O) = 0 , a

0,O

0.1

is t h e c o e f f i c i e n t o f t h e X 2

term i n t h e power series e x p a n s i o n o f H(X1 ,X2).

we see t h a t a

t h e r e e x i s t s a unique g,(X,)Ek[[X1]] 0.

S i n c e f,(X2) =

which we have assumed is non-zero.

= cl;

0.1

x,.

t

Note t h a t H ( 0 , O ) =

X 1 + fo(X2)Ek[[X,,X2]].

By d e f i n i t i o n , a

= 0.

c, o n t o MA

+

1

n

1n-1

'nX2

By Theorem 7.70,

such t h a t g,(O) = 0 and H(Xl,g,(X1))

=

By Neumann's Theorem (7.221, g , ( q ) is a w e l l - d e f i n e d element i n F, f o r For a l l y = q + c,EM"

a l l qEM".

us e x a m i n e t h e image of M A f(M^

t

x,)

+

+ x o , under f .

is c o n t a i n e d i n M" + c,.

Ma

g, is contained i n

c,)

-

f,(g(q

+

i.e.,

( i ) holds.

c, ( 7 . 3 4 : l ) .

x,)

+

Let

x,.

A s remarked a b o v e , by (7.34:1),

S i m i l a r l y , t h e image of M" + c,, under

By t h e Lemma 7 . 4 1 ,

H(q,g,(q))

i s a well-

S i n c e H ( X 1 , g o ( X 1 ) ) = 0 , a n d s i n c e Theorem 7.41

d e f i n e d e l e m e n t o f F. holds, H(g,g,(q))

x,.

+

d e f i n e g ( y ) t o be g o ( q )

c,,

Thus, 0 = H ( q , g , ( q ) )

0.

= =

-

A3

t o ( i i ) , l e t XEM"

q

t

By ( i ) , f ( x )

injective; thus f ( x )

=

- c,;

f ( g ( q + c,))

=

+

x,,

=

-

q

+

f,fgo(q)) =

hence f ( g ( q

+

c,))

-

q

-

9

+

co:

and l e t y = f ( x ) ; t h e n YEM"

y = f(g(y)) = f(g(f(x1)).

BY Lemma 0,

f

+

is

f ( g ( f ( x ) ) ) implies x = g ( f ( x ) ) ; e s t a b l i s h i n g ( i i ) .

As t o ( i i i ) , i t f o l l o w s immediately from ( i ) a n d ( i l l . (5)

Let L

=

.F,K((C))

b e a s d e f i n e d i n (7.82:O).

Let V be t h e Hahn

v a l u a t i o n on L , having v a l u e g r o u p C, v a l u a t i o n r i n g 0 , a n d maximal i d e a l M.

8.01

A primer on a n a l y t i c f u n c t i o n s o f a s u r r e a l v a r i a b l e

Let f , ( X )

Let B be as d e f i n e d i n ( 7 . 8 2 ) ;

cn-Xn be i n L[[X]].

=

339

t h e n B i s t h e s m a l l e s t convex s u b g r o u p o f G s u c h t h a t each cnecK((B)); which f i e l d we w i l l d e f i n e t o b e k . of B i s G.

Let A be t h e c a n o n i c a l d i r e c t summand

T h e n , a s we saw i n S e c t i o n 7.82, L

=

Ek((A)).

Let V^ be t h e

Hahn v a l u a t i o n on L having v a l u e g r o u p A , v a l u a t i o n r i n g 0- a n d m a x i m a l ideal MA.

Let P

[xEL: { V ( x ) }

=

>

Then f , ( X ) ~ k " X l l ,

B}.

and P = M A .

Let

c, and x, be i n L .

Given XEP + x,, we may u s e t h e Main Theorem t o a s s u r e

us t h a t f ( x ) =

c;(x

-

x , ) " is a w e l l - d e f i n e d e l e m e n t o f L.

THEOREM 0.

i n t o P + c,.

(i)

f maps P + x,

(ii)

f is i n j e c t i v e .

Assume t h a t c , f 0 ; t h e n

( i i i ) There e x i s t s a unique g , ( Y ) ~ k [ [ Y l ] s u c h t h a t i f , f o r a l l y = q + c ,

in P

+

c o t we d e f i n e g ( y ) = g , ( q ) + x,;

element o f P

+

x,,

f o r which f ( g ( y ) )

(iv)

g ( f ( x ) ) = x, f o r a l l XEP + x,.

(v)

Thus, f maps M ^ + x, o n t o MA

+

=

t h e n g ( y ) is a w e l l - d e f i n e d y.

c,, a n d g maps M A

+

c, o n t o M A + x 0 .

Assume t h a t K is a n o r d e r e d f i e l d , a n d t h a t F h a s t h e l e x i c o g r a p h i c o r d e r on i t ; t h e n (vi)

f is o r d e r - p r e s e r v i n g i f c ,

PROOF.

>

0 , and o r d e r - r e v e r s i n g

( i ) f o l l o w s from ( 7 . 3 4 : 1 ) ,

from Lemma 2 , and ( v i ) from Lemma 1 .

if c ,