Foundations of Analysis over Surreal Number Fields (North-Holland Mathematics Studies)

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

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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




=

Since A is an o r d e r e d group we know t h a t

Suppose t h a t ( 6 ) h o l d s ; t h e n y (xy

Since g is

g.

, a n d we s e e t h a t x




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 t a ( 4 1 , f o r a l l a < A , showing t h a t x Conversely l e t y = x + b , f o r s a n e beB. Then u ( y - aa) = v ( ( x

-

2

y i s i n B.

- aa)

+ b)

-

m i n . ( t , v ( b ) ) = ta ( 1 . 6 1 : 2 ( i v ) ) , p r o v i n g Lemma 2. a LEMMA 3. tive t o

V,

( i ) Let ( a a ) a < a b e a pseudo-convergent

sequence i n G r e l a -

and l e t beG; t h e n ( b f aa)a

i s a Hahn v a l u a t i o n o n H ( r e s p . CHI; and t h a t ,

(i)

p

(ii)

for each i d , p-'([t,-))/p-'((t,=))

( i i i ) (Hi)iEI

i s i s a n o r p h i c t o Hi.

i s t h e s k e l e t o n of H w i t h r e s p e c t t o p .

It

Norman L. A l l i n g

52

1.63

(Here B may b e a p r o p e r

Let A be a s u b g r o u p of a n o r d e r e d g r o u p 8.

class.)

The o r d e r o n B i n d u c e s a n o r d e r on A , u n d e r w h i c h i t i s a n o r d e r e d A w i l l b e r e f e r r e d t o as

group.

an o r d e r e d s u b g r o u p of B , a n d B w i l l b e

r e f e r r e d t o a s a n e x t e n s i o n of A.

ext(A') (5)

=

[bcB: t h e r e e x i s t s

Let A ' be a convex s u b g r o u p of A.

s u c h t h a t ( b ( S. l a [ ) .

aEA'

e x t ( A ' ) i s a convex s u b g r o u p of B. C l e a r l y e x t ( A ' ) , which we w i l l d e f i n e t o be B ' ,

PROOF.

v a l i n B a n d c l e a r l y i t i s c l o s e d under s u b t r a c t i o n . d e f i n i t i o n there e x i s t a,, a l E A ' C l e a r l y Ib,

Let

5

+ b,I

p r o v i n g t h a t b, aEA

+

Ib,I

+

18 a n i n t e r -

Let b a r b , c B ' .

Ib,I 2 lao[

+

lalI

=

Ilaol

+

I a l l [ (1.60:3),

blEB'.

and l e t A ' = (xEA:

t h e r e e x i s t s nEN s u c h t h a t 1x1 S n l a l ) :

Then

t h e p r i n c i p a l convex s u b g r o u p of B g e n e r a t e d by a .

ext(A') is

ext(A')

PROOF.

By

s u c h t h a t l b o l 5 l a o l a n d Ib,I 5 l a , ] .

i . e . , l e t A ' be t h e p r i n c i p a l convex s u b g r o u p of A g e n e r a t e d by a .

(6)

Let

Then

=

{ x c B : t h e r e e x i s t s nEN s u c h t h a t 1x1 5 n l a l ) .

L e t a b e a Hahn v a l u a t i o n of A a n d l e t B be a Hahn v a l u a t i o n of 9. For each t E V C a ( A ) ,

ext(a

-1

( [ t , m ) ) )

a-'([t,m))

i s a p r i n c i p a l convex s u b g r o u p of B.

t h e form B - ' ( [ j ( t ) , - ) ) , VCB(B).

I f j maps VCcr(A) o n t o VC ( B ) ,

B

extensio n of -

Such g r o u p s are of



Define g

i t i s an o r d e r e d group.

0 i f ng

>

For each geG' t h e r e e x i s t s

0 i n A.

Under t h i s o r d e r i n g o n

In a d d i t i o n , i t i s an o r d e r e d v e c t o r s p a c e

o v e r Q : i . e . , a v e c t o r s p a c e o v e r Q and an o r d e r e d group s u c h t h a t f o r a l l gEG' and q E Q , t h e n g

>

0 and q

>

0 i m p l i e s t h a t qg

>

0.

Clearly

G I

is an

Archimedean e x t e n s i o n of A ( 1 . 6 3 ) .

In 1936 Reinhold Baer [ l l ] proved t h i s i n a more

HISTORICAL REMARK,

e l e m e n t a r y way, a s f o l l o w s . ( a d d i t i v e ) Abelian group.

Let C

For t h e moment, l e t A b e a t o r s i o n - f r e e =

He d e f i n e d an e q u i v a l e n c e r e l a t i o n

my.

A x N , and a

l e t ( x , m ) and ( y , n ) be i n C.

on C a s f o l l o w s : ( x , m )

a

( y , n ) i f nx

Addition i s d e f i n e d a s f o l l o w s : ( x , m ) + ( y , n ) = (nx + my, m n ) .

b e d e f i n e d t o be C/a.

Then G i s an Abelian group.

=

Let G

F u r t h e r , t h e map acA

+

( ( a , l ) / a ) c G i s a monomorphism of A i n t o C , w h i c h we w i l l r e g a r d a s a n identification.

L e t ueZ a n d l e t V E N .

(ux,vm); then C i s a v e c t o r s p a c e over Q.

Let ( u / v ) . ( x , m ) be d e f i n e d t o be

Furthermore, C i s t h e s m a l l e s t

54

Norman L . A l l i n g

subspace of i t s e l f t h a t c o n t a i n s A . o r d e r e d group.

Define (x,m)

o r d e r e d v e c t o r s p a c e over Q .

>

1.64

Assume, i n a d d i t i o n , t h a t A i s a n

0 if x

>

0.

Under t h i s o r d e r , G i s an

T h i s c o n s t r u c t i o n i s , of c o u r s e , v e r y much

l i k e t h e u s u a l c o n s t r u c t i o n of Q f r o m Z .

G is order-isanorphic t o the

o r d e r e d v e c t o r s p a c e GI, c o n s t r u c t e d above. L e t v be a Hahn v a l u a t i o n on C , and l e t T the skeleton of C .

=

VC (C).

Let ( C t ) t c T

be

Let H be t h e f u l l Hahn g r o u p g e n e r a t e d by ( G t ) t e T

(1.63).

For each t c T , l e t h v

-1

([t,-))/v-’((t,-)),

t

be t h e c a n o n i c a l homomorphism of v

-1

([t,m))

which we d e f i n e t o be Ct (1.61: Theorem 1 ) .

onto After

p o s s i b l y r e p l a c i n g G t with a n i s a n o r p h i c copy of i t i n (R,+), we may assume that ~

S i n c e C i s v e c t o r s p a c e o v e r Q, and ht i s Z - l i n e a r , we

E (G1 . 6~1 ) .

may g i v e Gt t h e s t r u c t u r e of a v e c t o r s p a c e o v e r Q i n s u c h a way t h a t ht i s

Q-linear.

C o n s i d e r t h e f o l l o w i n g s h o r t e x a c t sequence i n t h e c a t e g o r y of

Q-spaces and Q - l i n e a r maps:

S i n c e Q is a f i e l d , we know t h a t (0) s p l i t s : i . e . , t h e r e e x i s t s a Qlinear injection k

-1 t of Ct i n t o v ( [ t , - ) ) such t h a t h t * k t i s ’ t h e i d e n t i t y

map of C t , and s u c h t h a t f o r a l l g e v - ’ ( [ t , - ) ) , Thus we c a n write v of Ct and v

(1)

-1

((t,m))

-1

([t,m))

(1.60).

(g

-

kt(ht(g))Ev

-1

( ( t , m ) ) .

as 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 Note t h a t

t h e c o n s t r u c t i o n of kt u s e s t h e axiom of c h o i c e .

1.64

55

Preliminaries

-1

Let e t = k t ( l ) ( c v

t ' l e t r e t be

For a l l rcG

( [ t , m ) ) ) , f o r each t E T .

d e f i n e d t o be k t ( r ) .

For g&*,

LEMMA 0 .

h (g - re

PROOF.

t

l e t v(g)

-

(g

0,

=

t

t and h t ( g )

=

ret)€"

-1

r.

=

- ret) >

u(g

-

( ( t , - ) ) , and v ( g

v(g).

>

ret)

t.

For each t c T , l e t f t be t h e e l e m e n t of H whose s u p p o r t i s It] a n d such t h a t f t ( t )

=

1.

Assume t h a t t h e r e e x i s t s a p r o p e r

[45]).

M A I N LEMMA (Hausner-Wendel

s u b s p a c e Go of G t h a t c o n t a i n s [ r e t t:

tcT and r cG 1, and assume t h a t a map

t

t

F, of C, i n t o H has been d e f i n e d having t h e f o l l o w i n g p r o p e r t i e s : F, i s Q - l i n e a r ;

(i)

( i i ) f o r each

and f o r each rcGt, F , ( r e

tET

=

t

rft;

( i i i ) F, i s i n j e c t i v e ;

f o r e a c h f E F ( G , ) , and e a c h c u t C

(iv)

where C f ( t )

f ( t ) for

=

(v)

f o r a l l gcG,, u ( F , ( g ) )

(vi)

F, p r e s e r v e s

Let XEG

-

and C f ( t )

tEL, =

( L , R ) of T , t h e n CfEF(C,),

=

=

0 f o r tcR;

v ( g ) ; and

t .

=

[v(x

-

number A , and a n i n j e c t i o n a c W ( h )

+

T h e r e e x i s t s a non-zero l i m i t o r d i n a l a EG,

such t h a t acW(h)

is o r d e r - p r e s e r v i n g map, f o r which [ t : a a

(B) rnin.{v(a

a

Let -

XI,

Let ~ E G ,

A s a c o n s e q u e n c e we s e e

Since r e € G o , y + r e t c G , . t

t h a t T o h a s no g r e a t e s t e l e m e n t .

taET,

Clearly To f 0.

y ) : YEC,].

A s we saw i n Lemma 0 , t h e r e exists rcC t such t h a t

< 6
-y'.

=

> -y.

0 then 0

If q

>

0; then x

>

-y/q.


y (resp. x < y) implies x' > y' ( r e s p . x' < y ' ) . P r o c e e d i n g by c o n t r a d i c t i o n , a s s u m e t h a t x > y and x' < y ' . Let t = p(y' - X I ) . S i n c e t h e o r d e r o n H is t h e l e x i c o g r a p h i c o r d e r , we know t h a t x ' ( t ) < y ' ( t ) , a n d we know t h a t x ' ( t 1 I ) = y ' ( t " ) , f o r a l l t"ET w i t h t " < t . Assume f o r a moment, ( a ) t h a t t > 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


q n ,

>

q,

4.05

>

and w

... ,

B1,

Bn > 0.

R e g a r d i n g s u c h s u m s as f o r m a l power s e r i e s a l l o w s u s t o a d d a n d

Let u s a l s o r e g a r d t h e o r d i n a l number 0 as t h e empty expan-

m u l t i p l y them.

s i o n ( 0 1 , a n d l e t i t b e a d d e d a n d m u l t i p l i e d a s a formal power series. Such sums and p r o d u c t s g i v e r i s e t o t h e n a t u r a l numbers.

a n d p r o d u c t of o r d i n a l

C55, pp. 246-2611.)

(See e.g.,

Note t h a t we have y e t t o p r o v e t h a t x we know t h a t i t is a game ( 4 . 0 4 ) .

t i o n of a d d i t i o n between games.

+

y i s a l w a y s a number; however

F u r t h e r , we may r e g a r d (0) as a d e f i n i -

As s u c h we c a n e s t a b l i s h p r o p e r t i e s a b o u t

F i r s t note that

it.

(3)

0

+

x

=

PROOF.

x , f o r a l l XENO. Assume t h a t 0

+

u = u is t r u e f o r a l l UEO

U’

and l e t b ( x ) = a.

(Note that i f a

=

0, t h e i n d u c t i o n h y p o t h e s i s i s empty, x

noted i n ( 2 ) , 0

+

x

+

xR I .

Since 0

x.)

By d e f i n i t i o n , 0

+

x

(0

=

+

xLIO

L R h y p o t h e s i s ) i s {x Ix 1, which i s x .

x

(4)

+

y

=

PROOF.

+

x = {OL

+

x, 0

+

L R x 10

+

x, 0

( 1 1 , t h e r e a r e n o 0L , o r 0 R : i . e . , 0 h a s n o o p t i o n s

=

Thus 0

(4.00).

=

0 , a n d , as

=

+

R x 1 , which ( u s i n g o u r i n d u c t i o n

o

y + x , f o r a l l x , YENO. Let x a n d y be chosen s o t h a t f o r a l l U , V E N O ,

n a t u r a l sum, b ( u )

+

(Note t h a t i f

b ( v ) , i s less t h a n b ( x )

f o r which t h e

b(y) = a; then u

v

v

=

(Y

=

0, t h e n t h i s i s t h e empty i n d u c t i o n h y p o t h e s i s ,

x

y , and hence x

+

y

+

+

+

u.

= 0 =

x.)

S i n c e any o p t i o n z of x ( r e s p . y ) i s s i m p l e r

than x (resp. y): i.e., b ( z )

< b ( x ) ( r e s p . b ( z ) < b ( y ) ) , we c a n u s e t h e

=

y

+

i n d u c t i o n h y p o t h e s i s t o show t h a t x [y

t

L

x , y

L + x I y + x R , yR

+ XI

=

y

+

y

+

x.

=

IxL 0

+

y, x

+

yLlxR

+

y,

x

+

yR)

=

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

4.05

135

Let u s now c o n s i d e r two s t a t e m e n t s which we w i l l p r o v e f o r a l l u, v ,

a n d z i n No.

w,

The f i r s t o f t h e s e , we w i l l c a l l P ( u , v : w , z )

is the

following: ( i ) u 2 v a n d ( i i ) w 5 z, implies ( i i i ) u

(5)

w 2 v

+

(iv) Strict

z.

+

i n e q u a l i t y i n ( i ) or ( i i ) i m p l i e s s t r i c t i n e q u a l i t y i n ( i i i ) . O u r i n d u c t i o n w i l l be w i t h r e s p e c t t o rnax.(b(u) + b ( v ) , b(w) + b ( z ) ) , Let N ( x , y ) b e t h e s t a t e m e n t t h a t

which we w i l l d e f i n e t o be b ( P ( u , v : w , z ) ) .

x

(6)

+

L

y i s a number: i . e . , t h a t {x

+

Let b ( N ( x , y ) ) be d e f i n e d t o be b ( x )

y, x +

+

L

y 1

< {xR

t

R

y , x + y }.

b ( y ) , and consider t h e f o l l o w -

ing statement:. ( 5 ) a n d ( 6 ) h o l d , f o r a l l u, v , w , z , x , a n d y i n No.

(7)

PROOF.

To e s t a b l i s h ( 7 ) , we w i l l p r o c e e d b y i n d u c t i o n o n

max.(b(P(u,v:w,z)),

b ( N ( x , y ) ) ) , w h i c h we w i l l c a l l b ( Q ( u , v : w , z : x , y ) ) .

(Note t h a t i f b ( Q ( u , v : w . z : x , y ) ) Let a

(5) and ( 6 ) a r e t r u e . )

>

= 0,

then u

=

v = w

=

z =

x

=

y

=

0, and

0 , a n d assume t h a t f o r a l l u, v , w , z , x ,




xy.

xyL

-

xRyL 2 xR'y + xyL'

-

xR'yL'

> xy.

f o r e a c h xL, t h e r e e x i s t s xL' 2 x L ; a n d f o r each y L t h e r e

By ( O ) ,

L'

xLyR h xL'y

xL' a n d y R ' f o r w h i c h

4).

PROOF of

exists y

-

xy.

F o r each xR a n d yL t h e r e e x i s t x R ' a n d y L ' f o r w h i c h X Y +

(i)

xyR

- x R' y R '
xy t h u s xy

< xL ' y

exists Y

L'

+

2 Y

L

consequence, 0

>

R' x y

+

t h u s xy


(x -

-

2 xy

R'

)(y

-

y

L'

) 2 (x

-

.

R

; a n d f o r e a c h yL t h e r e

1 L (x

-

R

L

x ) ( y - y 1.

AS

a

R ' L' R xy - x y - x y L ' + x y L xy x y - xyL + x R y L , a n d L' R ' L' R L R L xy x y 5 x y + xy - x y i p r o v i n g ( 4 ) . R'

-

-

To c o m p l e t e t h e proof of ( 3 ) , we c a n r e a s o n as we d i d i n t h e proof of (1).

a

PROPERTIES OF DIVISION I N No

4.10

THEOREM. t h a t xy = 1 .

Were x then -x(z)

-

Let x b e i n No, w i t h x n o t 0.

T h e r e e x i s t s y i n No s u c h

Thus, No i s a n o r d e r e d f i e l d .




Assume t h a t x

0 in

No.

T h e r e e x i s t s a YENO s u c h t h a t xy = 1 .

I n t h i s s e c t i o n , t h e f o l l o w i n g r e s u l t w i l l a l s o b e of u s e . (1)

T h e r e e x i s t s u b s e t s L and R of t h e s e t of p o s i t i v e elements of No such t h a t L




Since x

timely (4.02).

[x

L

0 , and s i n c e 0

Let L

t h a t 0 5 xL ( 4 . 0 2 : 1 4 ) .

=

=

L

[ x : xL

R x 1 , the r e p r e s e n t a t i o n being

I

[I},

=

155

t h e r e must e x i s t a n xL s u c h

> 01, and

let R

=

[x

R

1.

Then x

=

[O,LIR}; e s t a b l i s h i n g ( 1 ) .

( 1 ) We may write x a s [ O , x L l x R } , w i t h e a c h

Conway d e f i n e d t h e m u l t i p l i c a t i v e i n v e r s e y of x as f o l l o w s

being timely. [24,

xL > 0 , t h e r e p r e s e n t a t i o n

p . 211.

(1

+

(xL

L L x)y ) / x , ( 1

-

+

(xR

-

R R x)y ) / x I.

Conway writes t h e f o l l o w i n g a f t e r t h i s d e f i n i t i o n .

"Note t h a t ex-

p r e s s i o n s i n v o l v i n g yL and yR a p p e a r i n t h e d e f i n i t i o n o f y . t h a t r e q u i r e s u s t o ttexplaintt t h e d e f i n i t i o n .

It is this

The e x p l a n a t i o n i s t h a t we

r e g a r d t h e s e p a r t s of t h e d e f i n i t i o n as d e f i n i n g new o p t i o n s f o r y i n terms So even t h e d e f i n i t i o n of t h i s y is an i n d u c t i v e one. t [ T h i s

of o l d o n e s .

i s i n a d d i t i o n t o t h e t t o t h e r t l i n d u c t i o n s by which we s u p p o s e t h a t i n v e r s e s L

for the x

and x

R

have a l r e a d y been f 0 u n d . 1 ~ ~C24, p . 21.1

In a footnote t

[24, p . 211 Conway g i v e s t h e f o l l o w i n g v e r y i n s t r u c t i v e example. Example 0. as [0,21}.

Let x R

=

3.

Thus x c a n be w r i t t e n , as i t is above i n ( l ) ,

.

Hence [ x ) i s empty, and xL

=

2.

By (21, t h e g e n e r a l f o r m u l a

for y is

Let yoL

=

0.

Note t h a t yoL

BY ( 3 ) , we s e e t h a t y I R

=

Let us make no c h o i c e of a y o

S i n c e t h e r e i s no y o R , t h e r e i s n o y 1

1/2.

U s i n g ( 3 ) , we s e e t h a t y Z L = 1 / 4 . y S R = 11/32, y G L = 21/64, y,R

< 1/3.

=

T h i s t h e n y i e l d s y,R = 318, y,L

43/128,

...

=

R

L

. .

5/16,

Writing t h i s i n t h e customary

Norman L. A l l i n g

156

way g i v e s y = ( 0 , 1/4, 5/16, 21/64,

... I

4.10

1/2,

3/8,

11/32,

The decimal v e r s i o n of t h i s i s even more i n s t r u c t i v e . {O,

0.25,

0.3125,

0,328125,

... I

0.5,

0.375,

0.34375,

431128,

...

I.

It is t h e following: 0.3359375,

... ) .

I n t r y i n g t o u n d e r s t a n d ( 2 ) more f u l l y , n o t e t h a t e x c e p t f o r 0 t h e o p t i o n s g i v e n i n (2) a r e d e t e r m i n e d i n p a r t by t h e c h o i c e of w h i c h o p t i o n o n e c h o o s e s t o c o n s i d e r , r i g h t or l e f t .

Let u s make t h e f o l l o w i n g

def i ni t i o n s .

(4)

(i)

R

-

x)y ) / x ,

be d e f i n e d t o be ( 1 + ( x L

-

x)y ) / x ,

Let [R:Ll(y) b e d e f i n e d t o be ( 1 + ( x

( i i ) l e t [L:R](y)

L

R

R

L

( i i i ) l e t [ L : L l ( y ) b e d e f i n e d t o be ( 1

+

( x L - x ) y L ) / x L , and

l e t [R:Rl(y) b e d e f i n e d t o be ( 1

+

(xR

(iv)

-

R R x)y )/x

.

Then, a c c o r d i n g t o (21,

PROOF of ( 0 ) .

-

Note t h a t (0) i s t r u e i f x




is.

(4)

D e f i n e z t o be [x

(5)

22 = z + z =

(6)

z

+

x

-

2-"


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, 0 and Y > 0 .

uy.

(4)

X

=

{O,aw

X

L bw

X

b(y), let X

=

w

X

and l e t Y

By d e f i n i t i o n ,

L

R

+

171

R

] and Y = (0,cwy IdaY ) , where a , b , c , d a r e i n R t .

Then XY =

L

L

(5)

10. awx

(6)

L (0, a w x ty

+'

+

+

cw '+Y

cw '+Y

L

L

-

acw

-

acw

L

R

L

ty

ty

~

L

ty

bw

, bw

R

+'

R

t

dw"Y

t

dw"y

R

-

bdwX ty

R

R

I

-

R R bdwX ty

I

S i n c e ( 5 ) and ( 6 ) a r e i d e n t i c a l , t h e Theorem is proved. Applying t h e Theorem, we o b t a i n t h e f o l l o w i n g c o r o l l a r y :

(7)

For a l l XENO, w

-X

=

l/w

X

.

F u r t h e r , t h e map t h a t takes XENO t o w x i s

a n o r d e r - p r e s e r v i n g homomorphism o n t o t h e class A of a l l l e a d e r s i n

No; t h u s A i s an o r d e r e d group, under m u l t i p l i c a t i o n , t h a t is o r d e r l s u n o r p h i c t o t h e a d d i t i v e group (No,+) of No.

4.41

FINITE LINEAR COMBINATIONS OF w - x ( 1) ,

... , w - x ( n )

OVER R

We w i l l be concerned i n t h i s s e c t i o n w i t h f i n i t e l i n e a r combinations

Norman L . A l l i n g

172

of elements of t h e form w Y o v e r R.

4.41

Let aeR, and l e t Y E N O .

The f o l l o w i n g

a r e obvious.

(0)

(i)b 0.~')

= 0,

0

f o r all YENO, and ( i i ) b ( a w 1 = b ( a ) , f o r all aeR.

Since b ( - x ) = b(x), and having c o n s i d e r e d ( O , ( i ) ) ,

L

f i e l d of Mu.

R

Since b ( a ) 4

Let a = { a la J E R .

W,

w e know ( b y c o n v e n t i o n

R L t h a t ( a L , a ) i s a t i m e l y r e p r e s e n t a t i o n f o r a: t h u s each a

(4.02:15)),

and each a

F u r t h e r , w e saw i n (4.30) t h a t R i s a sub-

Ow.

=

0.

A s we

As b e f o r e , l e t D denote t h e r i n g of a l l dyadic numbers ( 4 . 2 1 ) .

saw i n S e c t i o n 4 . 2 1 , D

>

assume t h a t a

R . is i n D.

Since a { aL wY

LEMMA.

awy

PROOF.

Let {a

=

L'

I

>

0 , we may a l s o assume t h a t each aL

0.

aR wY I .

{dcD: 0

=

>


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




y , t h e n l e t R**

(L**,R**)

-

which i s a b s u r d .

R*.




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),



0 ) ) ( b y Theorem 4.40).

6.50

Notice f u r t h e r m o r e , t h a t t h e map

V ( W - ~ ) E N O i s t h e i d e n t i t y map of No ( 6 , 1 0 : 3 , i ) ;

t h u s V is a n o r d e r -

p r e s e r v i n g isomorphism of E o n t o t h e v a l u e group of V . I n S e c t i o n 6.43 we began t o develop Conway's i d e a of t h e

Let X E N O * .

normal form f o r x , b u t broke o f f our d i s c u s s i o n a f t e r p r o c e e d i n g a f i n i t e number of s t e p s , i n o r d e r t o develop t h e i d e a of an a-power series i n No. Having developed these i d e a s i n S e c t i o n 6 . 4 3 , l e t us s y n t h e s i z e t h e t w o ideas i n t h e following.

Assume t h a t f o r some A i n On, an w-power s e r i e s S A h a s been g i v e n s u c h ' t h a t , f o r a l l 0 S A ,

be c a l l e d t h e

Let a w

a-term

V(x

-

-- la

.

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.

t h e n caA

>

>

0.

0.

Let a ^ be t h e l e a s t e l e m e n t

Let b ^ b e t h e l e a s t e l e m e n t i n

S i n c e G i s t h e o r d e r e d d i r e c t sum of A a n d B , a n d

31 4

Norman L . A l l i n g

since j ( x , )

c

=

a,b

t a + b ,a ^

~ > 0 , ~j ( x , ), > 0.~

Since c

7.81

+

bA

7.80

i s t h e l e a s t element of s u p p ( j ( x , 1).

~

DECOMPOSITION OF CERTAIN POWER SERIES FIELDS

Let T be a non-empty ordered s e t , a n d l e t (GtltET be a f a m i l y of non-

z e r o , Archimedean o r d e r e d g r o u p s . Let H be t h e f u l l Hahn g r o u p g e n e r a t e d by ( G t ) t r T ( 1 . 6 3 1 , a n d l e t C = H ( r e s p . E H ) ( 1 . 6 3 : l , i i i ) . L e t B be a convex subgroup of G .

Let v be t h e Hahn v a l u a t i o n o f C ( 1 . 6 3 ) ;

then v(B)

i s a n u p p e r - s a t u r a t e d s u b s e t TB# (1.30) of t h e e x t e n d e d v a l u e s e t T# of v

(1.61).

Let TAN = ( t s T C : ( t ] < TB#]; t h e n i t is a l o w e r - s a t u r a t e d s u b s e t

of T I .

Let A

=

(hc5H: s u p p ( a ) is a s u b s e t of TA%l. Then, as we saw i n

S e c t i o n 1.65, C is t h e o r d e r e d d i r e c t sum of A and B (1.60). Let K be a f i e l d . (resp. EK((C))).

( r e s p . f,F,((A))).

Let F,

-

Let F be t h e formal power s e r i e s f i e l d K ( ( G ) )

K((B))

isanorphism j of F , o n t o F.

t i o n of -

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

and l e t F,

=

F,((A))

I n S e c t i o n 7.80 we saw t h a t there e x i s t s a n a t u r a l KT h i s w i l l be c a l l e d a power series d e c o m p o s i -

F.

What a decomposition a c c o m p l i s h e s is t h a t i t a l l o w s u s t o w r i t e F as

a f i e l d of f o r m a l power s e r i e s o v e r a f i e l d of f o r m a l p o s e r series. 7.82

THE M A I N THEOREM

B e c a u s e o f t h e c a n o n i c a l f o r m a l power series s t r u c t u r e of ENo and

ECx, we w i l l be able, w i t h o u t m o d i f i c a t i o n , t o a p p l y t h e Main Theorem t o these fields. I n o r d e r t o p r o v e t h e Theorem f o r a wider r a n g e of examples, l e t u s proceed as f o l l o w s . (0)

(i)

Let E be a p o s i t i v e r e g u l a r i n d e x ,

(ii)

l e t T be an ordered s e t whose l o w e r c h a r a c t e r is a t l e a s t

(iii) let

w

E' be a f a m i l y of non-zero Archimedean o r d e r e d g r o u p s ,

7 .a2

Power series : formal and hyper-convergent l e t H b e t h e f u l l Hahn g r o u p o f ( G t I t E T ,

(iv) (v)

l e t K be a f i e l d ,

(vi)

a n d l e t L = SK(

(See, e.g.,

and l e t G = gH,

f o r a r e f e r e n c e t o t h e meaning of ( i ) , ( 1 . 3 0 )

(1.30:3

f o r ( i i ) , ( 1 . 6 0 ) f o r ( i i i ) , ( 1 . 6 3 ) f o r ( i v ) , and ( 6 . 2 2 ) for t h e n o t a t i o n of (7.401,

...

1

IkmO l s u m ( v ) = k C(v).X1

=

vi).)

Recall

l e t neN, and l e t V

F(X)

315

L e t hcZ(>O) a n d l e t u s d e f i n e F(X)

"n a x n

h t o be

... , X n l l .

be i n LCEX,,

lkIhIsum(v)=k C(v) *xv.

S i n c e e a c h C(v) is i n L, C(v) may be w r i t t e n as f o l l o w s :

c ( v ) - t g ( a ' V ) , with c ~ ( v ) E K ,g ( a , v ) e G , and h ( a ) a

Iu

B(IIhI;

then P ( I l h is a

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

7.04

Assume t h a t 111

LEMMA.




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




0 ( r e s p . cm < 0 ) ;

t h e n f is a c o n t i n u o u s , C-continuous, a n a l y t i c , O n - a n a l y t i c , i n j e c t i o n o f

P

+

x, o n t o P

+

c,;

which p r e s e r v e r s ( r e s p . r e v e r s e s ) o r d e r .

A p r i m e r o n a n a l y t i c f u n c t i o n s of a s u r r e a l v a r i a b l e

8.05

349

( i i ) Assume t h a t m is e v e n a n d t h a t cm > 0 ( r e s p . cm < 0 ) ; t h e n f i s

a continuous, €-continuous, order-preserving ( r e s p . order-reversing) t i o n of P(LO)

+

( r e s p . o r d e r - p r e s e r v i n g ) i n j e c t i o n of P ( S 0 ) + x,

and a n o r d e r - r e v e r s i n g o n t o P(L0)

+

injec-

x, o n t o P(L0) + c, ( r e s p . of P(L0) + x, o n t o P ( S 0 ) + c , ) ;

c, ( r e s p . of P ( S 0 )

+ c,

o n t o P(S0) + c , ) ; which is, con-

t i n u o u s , € - c o n t i n u o u s , a n a l y t i c a n d O n - a n a l y t i c o v e r P + x,. PROOF.

Let x

=

p

+

x 0 s P + x , , and n o t e ( 0 ) t h a t f(x)

- c,

=

f,(p)

=

c . ( g , ( ~ ) ) ~ .T h u s , f o i s t h e c o m p o s i t i o n of g o , ( d i s c u s s e d i n ( I ) ) , and m

x ~ g N o-+ xm ( d i s c u s s e d i n (211, m u l t i p l i e d by cm; t h u s t h e o r d e r p r o p e r t i e s From t h e s e we s e e t h a t the two k i n d s of

a s s e r t e d i n t h e Theorem h o l d .

c o n t i n u i t y , as d e s c r i b e d i n t h e Theorem, h o l d . t h a t f is On-analytic;

lnzocn.(x -

be

x,)

n

Concerning t h e a s s e r t i o n s

t h i s follows from t h e f a c t t h a t f ( x ) was d e f i n e d t o

, for a l l

XEP +

x,, and from Theorem 8.00.

o

8.1 0 LOCAL PROPERTIES OF POWER SERIES I N SEVERAL VARIABLES Assume t h a t L i s a s g i v e n i n S e c t i o n 7.82, and l e t V

(O)

FO(X)

=

l s u m ( v ) = h A v).X1

lh:,

... s X n

1

V

n

be i n L"X,,

Let Pn be t h e prime polyd s k of hyper-convergence of F,.

Theorem, we know t h a t f o r a l l p is a w e l l - d e f i n e d e l e m e n t i n L. =

=

(x,, 'hz0

...

, xn)

(Isum(v)=h L e t A and B

Let

IT,

=

p

+

B,

(PI

9

"'

L e t x,

x,; t h u s , x - x,

A(v)(x

=

=

- x,)')

* pn)Epn, (xo,,,

=

=

pep".

... , X n l l . By t h e Main

( l s u m ( v ) = h A ( v ) p")

... , X ~ , ~ ) E L "a,n d

let x

Let A ( 0 ) = CEL; t h e n F ( x )

i s a well-defined element o f L.

b e d e f i n e d as t h e y have b e e n , s a y i n S e c t i o n 7.92.

and n 2 b e t h e c a n o n i c a l p r o j e c t i o n s o f G o n t o A a n d B r e s p e c t i v e l y .

By Neumann's Lemma ( 7 . 2 0 ) , g i v e n gEG t h e r e e x i s t s rn(n,(g))EZ(ZO), s u c h t h a t

Norman L. Alling

350

= h , v = 0 , v '1 1 2'

A(v)(x - x,)')

...

= h , v1=0, v2=0,

n lj=l (xj -

X ~ , ~ ) * ~ ~ (where X ) ,

,Vn-l=O,

g (x)EO*.

J

+

8.10

... +

vnLl

A ( v ) (x

- x,) V )

=

Thus we see t h a t

Let Ln b e given t h e product topology of t h e v a l u a t i o n topology on L.

S i n c e G is t h e ordered d i r e c t sum of A and B, A is c o f i n a l i n G : t h u s ( 1 proves LEMMA 0.

Limx+x,F(x)

= F(X,).

I n S e c t i o n 8 . 0 3 we showed t h a t Lemma 0 , of t h a t s e c t i o n , c o u l d be combined w i t h Theorem 7.92, t o prove Theorem 0 of S e c t i o n 8.03. Using t h e same l i n e of r e a s o n i n g , we may u s e Lemma 0 (above), and Theorem 7.93 t o prove THEOREM 0.

F is a continuous map of P" + x, i n t o P

+

c.

I n S e c t i o n 8 . 0 3 we a l s o c o n s i d e r e d d i f f e r e n t i a t i o n . Let us now c o n s i d e r j ' t h p a r t i a l d e r i v a t i v e s , 3F(x,)/3xj, of F e v a l u a t e d , a t x,. Let xi

-

x ~ , for ~ , all i C

lhI, A ( h X ( j ) ) ( X

-

X,)

j , and

let x

hXIJ1

I

(xj

j

- F(x,) = - ~ ~ , ~ A)( h*~ ( lj l )~( x-:x,)~ (h-1)xI.j) # x

0,j;

t h e n F(x)

where x ( j l is t h e characteristic f u n c t i o n of (jI d e f i n e d on I 1

Since

&,Il A ( h x I j f ) ( x - x,) ( h - l ) X ( j 'is i n On,

we see t h a t

,

.. .

9

, nl .

8.10

35 1

A primer on a n a l y t i c f u n c t i o n s of a s u r r e a l v a r i a b l e

C l e a r l y t h e second e x p r e s s i o n i n ( 2 ) e q u a l s t h e f o l l o w i n g : A ( x ( j 1 ) +

lhm2A ( h x ( j l ) ( x -

x,) ( h - 2 ) X ' J 1 i s i n O^, f o r each

jEIl,

... , n ] ,

Hence we have proved t h e f o l l o w i n g . LEMMA 1 ,

Let xi

=

x

0,i'

f o r a l l i f j , and l e t x

f x

j

-

0,j'

then

Proceeding a s we d i d i n S e c t i o n 8.03, l e t u s c o m b i n e Lemma 1 a n d Theorem 7.93. and o b t a i n

THEOREM 1 . P

+

( i ) F i s a d i f f e r e n t i a b l e f u n c t i o n from Pn

+

x, i n t o

c , whose j ' t h p a r t i a l d e r i v a t i v e a t x , i s t h e j ' t h f o r m a l p a r t i a l

derivative aF(x,)/ax f e r e n t i a b l e over P

J' n

evaluated a t x,. + x,.

Thus ( i i ) F i s i n f i n i t e l y d i f -

F i n a l l y , ( i i i ) f o r a l l xlcPn

+

x,, F has a

Taylor series e x p a n s i o n , given by p a r t i a l d e r i v a t i v e s , which i s e q u a l t o t h e formal Taylor expansion g i v e n i n Theorem 7.93.

This Page Intentionally Left Blank

35 3

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York, 1950). - S t r u k t u r e n , Math. Z . 158 (1978) 147-155.

64

Schwartz, N . ,

rl

65

Sikorski, R . ,

Remarks on Some Topological Spaces of High Power, Fund.

Math. 37 (1950) 125-136.

66

- - - - _ _ _ _ _,_ On --

a n O r d e r e d A l g e b r a i c F i e l d , Warsaw, Towarzyztwo

Naukowe Warszawskie 4 1 (1948) 69-96. 67

van d e r Waerden, B . L . ,

Modern A l g e b r a , v o l . I ( U n g e r , N e w York,

1949). 68

Z a r i s k i , 0. and S a m u e l , P . ,

Commutative Algebra, vol. I (van

Nostrand, P r i n c e t o n , 1958). 69

.......................... Nostrand, P r i n c e t o n , 1960).

,

Commutative A l g e b r a , v o l . I 1 ( v a n

359

INDEX

A

A*

(=

s e t of non-zero elements of a r i ng A )

1.1 ( = absolute value i n an ordered group), 1.60 1.1 ( = a n a l y t i c norm i n a surcomplex f i e l d ) , 7.10 1.1 ( = cardinal number o r power), 1.03 1 1 . 1 1 ( = norm i n R n 1, 3.00 AC (axiom of ch o i ce) , 1.00

Addition ( i n No), 4.05 Addition theorem ( f o r binomial c o e f f i c i e n t s ) , 7.33 A d d i t i v e subgroup

of a r i ng o r a f i e l d -1, 1.60

( ( a , + )

A f f i n e l i n e , 3.10

Affine n space, 3.00 a-term, 6.50 a t h approximation, 4.50

Analytic a t a p o i nt , 8.00 Analytic norm ( = l * l ) , 7.10 Analytic on U, 8.00 e q u ip o ten t) , 1.03

a

(=

a

(=

commensurate), 1.61

a

(=

order e q u iv al ent ) , 1.01

I

4.30, 4.40

Archimedean ordered group, 1.60 Archimedean complete, 1.63 Archimedean extension, 1.63 Archimedean (ordered group, r i ng o r f i e l d ) ) , 1.60 Arcsine (over c e r t a i n formal power s e r i e s f i e l d s ) , 7.50 Arctangent (over c e r t a i n formal power s e r i e s f i e l d s ) , 7.50 Artin-Schreier Theory, 1.71 Associative law f o r m ul t i pl i cat i on ( i n No), 4.08

Norman L. A l l i n g

360

B

(B) ( = b i r t h - o r d e r axiom), 4.60 b ( = b i r t h d a y map), 4.01, 4.02. 4.03, f.60

Ball a b o u t a p o i n t of r a d i u s g r e a t e r t h a n g, 7.62 Binomial c o e f f i c i e n t s , 7.32 Binomial c o e f f i c i e n t s ( g e n e r a l i z e d ) , 7.93 Binomial series, 7.33 B i r t h - o r d e r axiom, 4.60 B i r t h - o r d e r f u n c t i o n , 4 . 0 1 , 4.02, 4.03, 4.60 B i r t h d a y , 4.01, 402 Born on day 0 , 4.00

(*I*), [.I

4.00,

4.01, 4.02.

4.03, 4.60

( = t h e convex subgroup g e n e r a t e d by

-1, 7.21

Breadth (of a pseudo-convergent s e q u e n c e ) , 1 . 6 2 , 6.40 C

C

( = Kuratowski c l o s u r e of

-1, 2.02

Canonical ( n a t u r e of power s e r i e s s t r u c t u r e o n No), 6.57 C a n t o r ' s normal form, 4.04, 6 . 4 3 C a r d i n a l (number, CC(.)

(=

=

power), 1 . 0 3

s e t of a l l Conway c u t s i n -1, 4.61

CD(.) ( = s e t of a l l C u e s t a Dutari c u t s i n * ) , 4.01, 4.02 Change s i g n ( a p o l y n o m i a l ) , 1.72

Cuesta D u t a r i c o m p l e t i o n o f -1, 4.02 Class of surreal numbers (No), 4.03 x ( - ) (=

Class of s u r r e a l numbers of h e i g h t 6, 4.03 C l a s s t h e o r y , 1.00 Closed c l a s s ( i n t h e i n t e r v a l t o p o l o g y ) , 1 .10 Closed s e m i - a l g e b r a i c s e t , 3.00 C l o s e d , s e m i - a l g e b r a i c s e t , 3.00 Closure operator

+

C

,

2.02

C o f i n a l ( s u b c l a s s i n a n o r d e r e d c l a s s ) , 1.30 C o i n i t i a l ( s u b c l a s s i n a n o r d e r e d c l a s s ) , 1.30 Commensurate ( e l e m e n t s i n a n o r d e r e d g r o u p ,

a),

4.40

Index Common p r e d e c e s s o r , 4.52 Compact o r d e r e d s p a c e s , 1.50 Complete ( = Dedekind-complete), 1.20 Complete o r d e r e d g r o u p s , 1. 66 C onj ugat e ( r o o t s ) , 1.72 Connected, 1 .20 C ont i nue t o change s i g n ( i n D 1, 5.51

A

Convex ( s u b g r o u p ) , 1.60 Convex ( s u b s p a c e ) , 3.30 Conway c u t r e p r e s e n t s , 4.02 Conway c u t s , 1.20 Conway's Normal Form, 6. 50 Conway's S i m p l i c i t y Theorem, 4.02, 4.03, 4.60 C os ecant ( o v e r c e r t a i n form a l power series f i e l d s ) , 7.50 Cosine ( o v e r c e r t a i n form a l power series f i e l d s ) , 7.50 C os i ne ( = ext ende d cosine f u n c t i o n o v e r E,Cx), 7.51

C r i t i c a l p o i n t , 1.74 C ues t a D u t a r i com ple tion ( = C D ( . ) ) , 4.02 C ues t a Dut ari c u t , 1. 20 C ues t a D u t a r i c u t r e p r e s e n t a t i o n ( o f a p o i n t ) , 4.02, 4.03 Cut p o i n t , 1.20 c u t s , 1.20

D D ( = d y a d i c numbers), 4.21 +

ad

( = wg-additive c l o s u r e o p e r a t o r ) , 2.02

a / a x i , 7.60 Decomposition ( o f f o r m a l power series f i e l d s ) , 7.81 Dedekind-complete,

1.20

Dedekind c u t , 1.20 Degree

m

( p o l y n o m i a l ) , 7.70

Degree ( o f V E Z ( > O ) ~ ) , 7.40 Dense i n i t s e l f , 1.10 Dense ( s u b c l a s s of a n o r d e r e d c l a s s ) , 1. 10

362

Norman L. A l l i n g

Dedekind-completion of O x , 5.50 D e r i v a t i v e (of a formal power s e r i e s ) , 7.60 Disconnected, 1 .20 Distance f u n c t i o n ( d ( - , . ) =

1.

*I

-

i n a surcomplex f i e l d ) , 7.10

D i s t i n g u i s h e d base, 2.01 D i s t i n g u i s h e d base of open sets, 2.01 D i s t i n g u i s h e d s u b b a s e , 2.01 D

x

(-

t h e s i m p l e s t Dedekind-completion of 0 ) , 5.50

x

Dyadic, 4.21 E

( E ) ( e t a axiom), 4.60

Embedding of q - f i e l d s , 1.75 F End p o i n t s (of a n i n t e r v a l ) , 1.10 € - t r a n s i t i v e , 1.02 Equipotent, 1.03 Equivalent ( s e t s ) , 4.02 Eta axiom, 4.60

q-character (- t r u e q-character) Q

5

,

1.40

- f i e l d , 0.03

11 -c ass, 1.40

E

EVCv

EVS

extended v a l u e c l a s s ) , 1.62

( 0

-

extended v a l u e s e t ) , 1.61

1 , 1 10

Exp

(s

extended e x p o n e n t i a l f u n c t i o n o v e r SCx)), 7.51

Exponential f u n c t i o n , 7.36 Exponential series, 7.36 Extended v a l u e c l a s s (EVC") of a group (C,vl with v a l u a t i o n , 1.62 Extended v a l u e set ( - EVS), 1.61 Extension ( o f a Conway c u t ) , 4.61 Extension ( o f a f i e l d with v a l u a t o n ) , 6.30 Extension (of an o r d e r e d s e t ) , 1 . 0 Extension ( o f a s e t ) , 8.00

Index

36 3

F

(F)

(=

f u l l n e s s a x i o m ) , 4.60

I F , < , b , B l , 4.03, 4.60 I F , < , b l , 4.03, 4.60 F a c t o r , 1.61 F a c t o r i a l ( g e n e r a l i z e d ) , 7.93 (FE)

(=

f u l l e t a a x i o m ) , 4.60

F i e l d of f o r m a l power series, 6.30

F i l l (a c u t ) , 4.02 F i n i t e i n t e r s e c t i o n property

(=

f.i.p.1,

2.30

F i n i t e o r d i n a l s , 1.02 F i r s t k i n d ( o r d i n a l s ) , 1.02 Formal i m p l i c i t f u n c t i o n theorem, 7.70, 7.71 Formal i m p l i c i t mapping theorem, 7.72 Formal i n v e r s e mapping theorem, 7.74 Formal power series ( i n n v a r i a b l e s ) , 7.40 Formal power s e r i e s o v e r f o r m a l power series f i e l d s , 7.80 Formally r e a l ( f i e l d ) , 1.70 F u l l ( c o n d i t i o n ) , 4.03 F u l l b i n a r y t r e e o f h e i g h t On, 4.50 F u l l e t a axiom

(=

F E ) , 4.60

F u l l f i e l d of f o r m a l power series, 6.30 F u l l Hahn group, 1.63 F u l l n e s s axiom

(=

F ) , 4.60 G

Game, 4.04 G a l a x i e s , 5.40 GCH ( = G e n e r a l i z e d Continuum H y p o t h e s i s ) , 1 . 3 0

G e n e r a l i z e d binomial c o e f f i c i e n t s , 7.93 G e n e r a l i z e d f a c t o r i a l s , 7.93 H

Hahn g r o u p , 1.63

Norman L. A l l i n g

364

Hahn v a l u a t i o n , 1.61

,

6.20

Hahn's Embedding Theorem, 1.64 Harzheim's Theorem, 4.02 Hausdorff ( s p a c e ) , 2.10, 7.62 Hausdorff's Normal

9 -type,

5

5.10

Height f u n c t i o n , 4.50

Hessenberg product ( = N a t u r a l p r o d u c t ) , 4.05 Hessenberg sum ( = N a t u r a l sum), 4.05 Hion's Lemma , 1.61 Htilder's Theorem, 1.60 ( = Hausdorff's normal n - t y p e ) , 5.10 5 F Hyper-convergent, 7.22

H

Ideal of i n f i n i t e s i m a l e l e m e n t s , 6.00 Ideal theory of a v a l u a t i o n r i n g , 6.70

I d e n t i t y Theorem ( f o r normal f o r m s ) , 6.51 Imaginary p a r t , 7.10 Immediate e x t e n s i o n , 1.63, 6.30 I m p l i c i t f u n c t i o n theorem ( f o r formal power series), 7.70, 7.71 I m p l i c i t f u n c t i o n theorem (for Neumann s e r i e s ) , 7.75 I m p l i c i t mapping theorem ( f o r f o r m a l power s e r i e s ) , 7.73 I m p l i c i t mapping theorem ( f o r Neumann s e r i e s ) , 7.75 ( I N ) ( - axiom: t h e r e is a s t r o n g l y i n a c c e s s i b l e c a r d i n a l number

Incomplete

(=

n o t Dedekind-complete),

1.20

Incomplete o r d e r e d groups, 1.66 Independence of r e p r e s e n t a t i o n (of Neumann s e r i e s ) , 7.83 I n f i n i t e l y l a r g e , 1.60 I n f i n i t e l y s m a l l , 1.60 I n f i n i t e s i m a l expansion (of a n a n a l y t i c f u n c t i o n ) , 7.65

*-, fa

1.10

( i n D A ) , 5.51

I n j e c t i o n ( = one-to-one map) I n j e c t i v e ( - being one-to-one)

.

I n t e r v a l , 1 10

1 1 , 1.00

Index I n te r v a l topology, 1 .10 I n te r v a l (c-closed), 2.12 I n t e r v a l (c-open), 2.12 Inverse mapping theorem ( f o r formal power s e r i e s ) , 7.75 Inverse mapping theorem ( f o r Neumann s e r i e s ) , 7.75 Is o late d I,

(= a

(=

convex subgroup), 1.60

strongly inaccessible car di nal number), 5.40

Jacobian ( matr ix) , 7.73

K K-valued c o e f f i c i e n t , 7.40 Kuratowski closure operator

(.

+

C

1, 2.02 1

Leader (of y ) , 4.40 Left c h a r acte r , 1 .30 Left-option, 4.00 Length (of a pseudo-convergent sequence), 6.41, 6.44

01, w i t h

an ordered group o r f i e l d )

P o i n t of s t a b i l i t y , 5.51 P o l e of o r d e r - n , 1.73 P o s i t i v e element

i n a o r d e r e d g r o u p , r i n g o r f i e l d ) , 1 . 6 0 , 1.70

Positive regular

ndex ( = p r i ) , 1 . 3 0

P o s s i b l y u n t i m e l y ( c u t r e p r e s e n t a t i o n ) , 4.09 Power (of a s e t = i t s c a r d i n a l number), 1.03

Index P r e d e c e s s o r , 4.50, 4.51, 4.53 Predecessor c u t r e p r e s e n t a t i o n , 4.54 Preserves

D e r i v a t i v e T e s t , 1.74 S e c t i o n (of an o r d e r e d c l a s s ) , 1.02 Semi-algebraic s e t , 3.00 Sequence ( i n a s e t ) , 7.21 S e t t h e o r y , 1.00 E ( = class of a l l s i g n e x p a n s i o n s ) , 4.50

No t o i t s s i g n expansions i n Z), 4.50 Sign-expansion ( f u n c t i o n 0 1 , 4.50 o ( = a map form

Simple d e n s i t y axiom ( = ( S D ) ) , 4.60 Simple z e r o , 1.73 Simpler

(=

of I b i r t h d a y ) , 4.01

Simplest Dedekind-completion, 5.50 S i n e ( o v e r c e r t a i n formal power series f i e l d s ) , 7.50 S i n e ( = extended s i n e f u n c t i o n o v e r ~ C X ) ,7.51 S i n g u l a r ( c a r d i n a l number), 1 .30 S k e l e t o n (of an o r d e r e d g r o u p ) , 1.63 S t a b l e v a l u e , 5.51

*

(=

.*

= IXE.:

x # 0))

S t r i c t l y d e c r e a s i n g ( s e q u e n c e ) , 7.21

Index

37 1

S t r i c t l y increasing (sequence), 7.21 Strictly-order-preserving (mapping), 1 . 0 1 , 1.60 Strictly-order-reversing (mapping), 1.01 S t r i c t l y p o s itive element ( i n an ordered group, ring o r f i e l d ) , 1.60, 1.70 Strong topology, 3.00 Strongly inaccessible ( car di nal number L), 5.40 Subsequence, 7.21

Subtraction ( i n No), 4.04 Successor, 4.50, 4.51, 4.53 n th e support of a vector .EM ) , 7.41 s u p p ( * ) ( = th e support of 1.63, 6.20 Support, 1.63, 6.20 supp(.)

(=

a ) ,

Surjection

(=

a map of one set onto another)

S u r j e c t i v e ( = a mapping t h a t is a s u r j e c t i o n )

Surcomplex number f i e l d s (Cx, and ~ C X ) ,7.10 Surreal monomorphism, 4.03 Surreal number f i e l d s (No, and CNo), 5.00 Sylow Theorems, 1.71 Symmetric, 4.21 T T,,

4.02

Tangent (over c e r t a i n formal power s e r i e s f i e l d s ) , 7.50 Tarski-Seidenberg Theorem, 3.00 Taylor-Neumann series, 7.91 The canonical d i r e c t summand, 1.65 The l i m i t (of a pseudo-convergent sequence), 1.64, 6.41, 6.42 Timely ( c u t r ep res ent at i on) , 4.02, 4.09 Topological f i e l d (under t he 6-topology), 3.40 Totally ordered s e t (=ordered s e t ) , 1.01 T ra n s f in ite inducti n , 1.02 Tree o r d er , 4.50, 4 51, 4.53 Triangle eq u alit y, .61 Triangle inequality 1.61 True n-character, 1 40

Norman L . A l l i n g

372

U U ( = group of

u n i t s of a v a l u a t i o n r i n g 01, 6.00

UCF ( u n i v e r s a l c h o i c e f u n c t i o n axiom), 1.00

U(g), 3.00 U n i v e r s a l l y embedding, 6.60 U n i v e r s e s ( i n s e t t h e o r y ) , 5.40 Upper c h a r a c t e r , 1 .30 Upper-saturated,

U^

1.30

( = U extended t o

Y), 8.00 V

V ( = v a l u a t i o n ) , 6.00

Value group, 6.00 Value s e t , 1.61 V a l u a t i o n r i n g , 6.00 V a l u a t i o n t o p o l o g y , 7.62 V a l u a t i o n t o p o l o g y and t h e i n t e r v a l t o p o l o g y , 7.63 VS ( = v a l u e s e t ) , 1.61

W

Weak c - t o p o l o g y , 2.01 Weakly i n a c c e s s i b l e ( c a r d i n a l number), 1.30 Well-ordered

( c l a s s ) , 1.02

W(g), 3.00

X x

f - t h e a t h approximation t o x ) , 4.50

E,B ( = t h e t;-topology g e n e r a t e d by a base B), 2.01 E,-closed,

2.01

c - c l o s e d s u b c l a s s e s of R n ,