Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra

268 Pages·2001·8.86 MB·English

by  Alexander Prestel, Charles N. Delzell (auth.)

Springer Monographs in Mathematics
Springer-Verlag Berlin Heidelberg GmbH
Alexander Prestel Charles N. Delzell
Positive Polynomials
From Hilbert's 17th Problem
to RealAlgebra
, Springer
AlexanderPrestel
Universitiit Konstanz
FachbereichMathematikund Statistik
Postfach 5560
78457 Konstanz
Germany
e-mail: [email protected]
CharlesN.Delzell
Louisiana State University
DepartmentofMathematics
Baton Rouge, Louisiana 70803
USA
e-mail:[email protected]
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Prestel,Alexander:
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CharlesN.Delzell.-Berlin:Heidelberg:NewYork:Barcelona;HongKong:London:Milan;Paris;
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Preface
Exactly 100 years ago, at the turn of the 19th to the 20th century, in his
famous address to the 1900International CongressofMathematicians,David
Hilbert [1900] presented a list of 23 problems that he considered to be the
most important problems left from the old century to be solved in the new
one. The 17th problem, in its simplest form, is as follows:
Suppose f E IR[X1,...,Xn] is a real polynomial in n indetermi
nates, and f(x) 2:: 0for allx E lR(n) . Does there then necessarily exist
a representation of f as a sum of squares of real rational functions,
i.e., in the form
for finitely manyr, from thefieldIR(X1,...,Xn) ofrationalfunctions
inX1,·.. ,Xn?
It did not take long for the problem to be solved:in [1926] E. Artin pre
sentedaquiteremarkablesolutionto the problem.Ratherthanconstructinga
representationoff asasum ofsquares ofrationalfunctions, Artinshowedthe
mere existence of such a representation, by an indirect proof. Nevertheless,
the solution offered a "global" characterization of positivity of polynomials
on lR(n).
This brings us to the main goal of our book: we seek characterizations
of those polynomials f that are positive on certain sets, themselves defined
by polynomial inequalities. In every case, these characterizations consist of
representing f within the ring ofall real polynomials in such a waythat the
required positivity of f is reflected instantly. Many results of this type have
been obtained over the last 75 years, all starting with Artin's solution of
Hilbert's 17th problem. New methods have been developed over the years,
focusing on "reality" and "positivity." In a sense, Artin's solution may be
understood as the beginning of "real algebra." Thus, not surprisingly, the
second goal ofthis book is to present an introduction to real algebra.
The book is based on a two-semester course having exactly these two
goals; it was given by the first author at the University of Konstanz during
the summer semester of 1999 and the winter semester of 1999-2000. The
present form of the book arose during a joint stay by both authors at the
VI Preface
Mathematical Research Institute in Oberwolfach (Germany), under its "Re
search in Pairs" program.
The part of the book that constitutes an introduction to real algebra
consists of:
Chapter 1, where weintroduce the theory ofordered fields and real closures
ofsuch fields (1.1-1.3);
Chapter 2, where wegive an introduction to semialgebraic sets and Tarski's
Transfer Principle (2.1-2.4);
Chapter 3, where wepresent a short introduction to the theory ofthe Witt
ring ofa fieldK, and study the total signaturemap on the space oforderings
of K-the "real spectrum" of K (3.1-3.3);
Chapter4,where weintroducetherealspectrumofan arbitrarycommutative
ring, and give a special descriptionofthe real spectrum ofthe particular ring
lR[X1,...,Xnl of real polynomials (4.1, 4.2, 4.4, 4.5); and
Chapter 5, where westudy rings in which every element is bounded on the
real spectrum, and give representations (i.e., homomorphisms) ofsuch rings
into rings of continuous real-valued functions on some compact Hausdorff
space (5.1-5.4).
Ourmaingoal-theimprovementsintherepresentationoff-is explained
in the Introduction, and pursued in Chapters 5 to 8. Artin's solution of
Hilbert's 17th problem is presented in Section 2.1 (Theorem 2.1.12). Gen
eralizations of this problem, as wellas improvements in the representation,
are found in Sections 3.5, 4.2,5.2, 5.3, 5.4, 6.3, 7.3, 8.3, and 8.4.
Each chapter has a section of exercises that may help the reader bet
ter understand what was treated in that chapter, and obtain some further
information. Finally, each chapter ends with "bibliographical and historical
comments," in which we try to inform the reader about the origins of the
notions and results in that chapter, and their connections to other work.
We are most grateful to Markus Schweighofer, a Ph.D. student at the
University of Konstanz, who contributed many ofthe exercises in the book.
He also carefully read all drafts ofthe book, offeringmany corrections, clar
ifications, and improvements.
Konstanz, Germany, October 2000 Alexander Presiel
Baton Rouge, U.S.A., October 2000 Charles N. Delzell
Please visit the book's web site, containing errata, updates, and other
material: http://www.math.lsu.edu/ro.Jdelzell/positive_updates.html.
And please send any corrections or suggestions that you may have to
alex.prestel<Ouni-konstanz.de or delzell<Omath.lsu.edu.
Contents
Introduction .................................................. 1
1. Real Fields 7
1.1 Ordered Fields. ........................................ 7
1.2 Extensions of Orderings ................................. 12
1.3 The Real Closure....................................... 16
1.4 Exercises ..... ..... ........ ... ........... .. .......... .. 24
1.5 Bibliographical and Historical Comments .................. 28
2. Semialgebraic Sets. ....................................... 31
2.1 Semialgebraic Sets...................................... 31
2.2 Ultraproducts ... ...... ......... .......... ... .......... . 36
2.3 Elimination of Quantifiers ............................... 41
2.4 The "Finiteness Theorem" " 45
2.5 Exercises.. ... .......... .. .... .... ...... .. .. ........ ... 47
2.6 Bibliographical and Historical Comments .................. 48
3. Quadratic Forms over Real Fields .. ... 53
3.1 Witt Decomposition .. .......... ........................ 53
3.2 The Witt Ring ofa Field ................................ 59
3.3 Signatures .. ..... ............ ............ .............. 62
3.4 Quadratic Forms Over Real Function Fields 68
3.5 Generalization of Hilbert's 17th Problem. ................. 74
3.6 Exercises ..... ......... ..... ......................... .. 77
3.7 Bibliographical and Historical Comments. ................. 79
4. Real Rings 81
4.1 The Real Spectrum ofa Commutative Ring................ 81
4.2 The Positivstellensatz ................................... 86
4.3 "Continuous" Representation ofPolynomials " 91
4.4 77a-Fields. .. .... .... ...... ........ ... ... ............. .. 94
4.5 The Real Spectrum oflR(X1, ,Xn) 101
4.6 Exercises 107
4.7 Bibliographical and Historical Comments 109
VIII Contents
5. Archimedean Rings 113
5.1 Quadratic Modules and Semiorderings 113
5.2 Rings with Archimedean Preorderings 119
5.3 Rings with Archimedean Quadratic Modules 124
5.4 Rings with Archimedean Preprimes 130
5.5 Exercises 134
5.6 Bibliographical and Historical Comments 136
6. Positive Polynomials on Semialgebraic Sets 139
6.1 Semiorderings and Weak Isotropy 139
6.2 Archimedean Quadratic Modules on IR[X1,•..,Xn] 142
6.3 Distinguished Representations of Positive Polynomials 145
6.4 Applications to the Moment Problem 152
6.5 Exercises 157
6.6 Bibliographical and Historical Comments 158
7. Sums of2mth Powers 161
7.1 Preorderings and Semiorderings ofLevel 2m 161
7.2 Semiorderings ofLevel2m on Fields 166
7.3 Archimedean Modules ofLevel2m 169
7.4 Exercises ........... .............. ...... ............. .. 176
7.5 Bibliographical and Historical Comments 177
8. Bounds 179
8.1 Length ofSums ofSquares 179
8.2 Existence ofDegree Bounds 183
8.3 Positive Polynomials over Non-Archimedean Fields 189
8.4 Distinguished Representations in the Non-Archimedean Case. 196
8.5 Exercise 201
8.6 Bibliographical and Historical Comments 201
Appendix: Valued Fields 203
A.1 Valuations 203
A.2 Algebraic Extensions 207
A.3 Henselian Fields 213
A.4 Complete Fields 223
A.5 Dependence and Composition of Valuations 230
A.6 Transcendental Extensions 236
A.7 Exercises ............. ..... ...... ... .... ........ ... .... 242
A.8 Bibliographical Comments ............................... 245
References. ................................................... 247
Glossary of Notations 255
Index 259
Introduction
Themostbasicnotionofreal algebraandreal analysis,in contrastto ordinary
algebra and complex analysis, is the notion of "positivity."
A subset T of a commutative ring A with 1 will be called a prepositive
cone or a preordering of A if
T+T~T, T·T~T, A2~T, and -l~T,
where A2 stands for the set ofsquaresofelements ofA.The set T enjoys the
basic properties of "positive" elements. Whenever such an object exists for
A, we call A a semireal ring.
The most prominent examples of such rings are:
(0.1) any subring A ofthefield lRofreal numbers, where T consists ofthose
elements of A that are nonnegative in lR, and
(0.2) the ring C(X,lR) of all continuous functions from a nonempty topo
logical space X to lR, where T consists of those functions f such that
f(x) 2: 0 for all x E X.
Iffor a preordering T we declare a :::; b to hold if and only if b- a E T
(a,bE A), then in (0.1) above weobtain a linear orderingon the underlying
set of A, i.e., one satisfying
a:::; b or b:::; a
for all a,bE A, while in (0.2), :::; need not be linear. Moreover, preorderings
:::; in general need not even satisfy
a :::; b, b:::; a => a =b.
From the definition of a preordering T, it is clear that every T contains the
L
set A2offinite sums ofsquares ofelements of A.IfA has any preordering
L =
at all, then clearly A2 is the smallest one. In the examples A lR and
A = C(X,lR) above, the set L A2 is a preordering;and in the first case it is
the only one.
The ring lR[X] := lR[XI ,...,Xn] of real polynomials in Xl,.. .,Xn may
=
be understood as a subringofC(lR(n),lR). For n 1,the preorderinginduced
L
by the canonical one in example 0.2 above is lR[X]2 (by the Fundamental
A. Prestel et al., Positive Polynomials
© Springer-Verlag Berlin Heidelberg 2001
2 Introduction
Theorem ofAlgebra).Ifn 2:: 2,however,the induced preorderingisno longer
L:JR[XF, as Hilbert had observed alreadyat the end ofthe 19th century. He
then conjectured, in the "17th problem" of his famous talk in Paris in 1900,
that the induced preordering might be
L
JR[X] n JR(X)2.
This means that every polynomial f EJR[X] that is positive semidefinite over
JR (i.e., f(a) 2:: 0 for alla E JR(n)) should berepresentable as asum of squares
ofrational functions in X.
In[1926] E.Artin proved that this isinfact so.Even though Artin's proof
appearsinacompletelydifferentlanguage, it pavedthe wayfortwoimportant
developments, the "real spectrum" and "Tarski's Transfer." Roughly speak
ing, Artin first added new "points" to JR(n) that forced f to be in I:JR(X)2
unless it was negative on a new point x. Then he proved that x could be
specialized to some old point a E JR(n) where f would remain negative; but
then f could not be positive semidefinite over JR.
Let us first explain how wewouldformulate Artin's proofnowadays.For
any real commutativering Awith 1,the realspectrumofA isdesignedsothat
A can be understood as a ring of functions on the real spectrum satisfying
the rule: "every positive semidefinite function isa sum ofsquares ofrational
functions." The "points" at which our functions should be evaluated are
obtained by looking at maximal preorderings P of A. Such maximal objects
have the additional property
P U -P =A and P n-P is a prime ideal.
Nowlet theset Sper A (the realspectrumofA) consist exactlyofpreorderings
1:
with this additional property. To every f E A one assigns a function
SperA -+ JR*, where JR* is a big ordered field containing all residue rings
A := A/(Pn-P),and such that the elements of P ~ A are positive in JR*.
Then j(P) := f +(pn-p) EJR*.Itisnowquite easyto provethat whenever
1
is positive semidefinite on SperA (i.e., j(P) 2:: 0 for all P ESperA), we
obtain tl,tz EI:A2 such that
(0.3)
for some e E N.l In the case of the polynomial ring A =IR[X], wetherefore
find
L
f = (f2e ~t2)tl E JR(X)2.
t
l
Thus iff ~ I:IR(X)2,there has to be some P ESperIR[X] with j(P) <o.
1 N= {a,1,...}.