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Positive polynomials. From Hilbert’s 17th problem to real algebra. (English) Zbl 0987.13016

Springer Monographs in Mathematics. Berlin: Springer. viii, 268 p. (2001).
Positivity of polynomials and other functions is one of the fundamental notions of real algebra. The classical question in this area is Hilbert’s problem:
Given a polynomial \(f\in\mathbb{R} [X_1,\dots, X_n]\) that is positive semidefinite on \(\mathbb{R}^n\), does \(f\) have a representation as a sum of squares in the quotient field \(\mathbb{R}(X_1, \dots,X_n)\) of \(\mathbb{R}[X_1, \dots, X_n]\)?
This question was answered by Artin (1926) in the affirmative. But there is a large number of variations of Hilbert’s problem that define very active areas of research to this day. The variations are concerned with: Other fields than the real numbers; other rings than polynomial rings; sum of squares representations in other rings than the quotient field; bounds in sum of squares representations: for the number of sums of squares, for the degrees and coefficients of the representing polynomials; positive definite functions on subsets of \(\mathbb{R}^n\), etc.
Starting from Hilbert’s problem, the authors present those parts of real algebra that are essential tools for the most recent contributions about positivity and that are relevant for a modern interpretation of the classical results. The topics of the first four chapters are real fields and real closures of ordered fields; semialgebraic sets and model theoretic methods, including elimination of quantifiers and the Tarski principle; quadratic forms over real fields and generalizations of the Hilbert problem and its solution; real rings, the real spectrum and a very general Positivstellensatz. Valuation theoretic methods are of great importance in real algebra. Basic facts about general valuations, as opposed to properties of valuations that are peculiar to real algebra, are collected in an appendix.
A question that sets the theme for much of the rest of the book is: If polynomials \(h_1,\dots,h_s\in\mathbb{R}[X_1,\dots,X_n]\) are given and if \(W_R(h_1, \dots,h_s)= \{x\in\mathbb{R}^n \mid h_1(x)>0, \dots,h_s (x)>0\}\) is the basic closed semialgebraic set defined by the \(h_i\), when is it true that a polynomial that is positive on the set \(W_R(h_1,\dots,h_s)\) has a representation of the form \(\sum_{\nu\in\{0,1\}^s}h_1^{\nu_1}\cdot\dots \cdot h_s^{\nu_s}\sigma_v\), where the \(\sigma_\nu\) are sums of squares in \(\mathbb{R}[X_1, \dots,X_n]\)? If such a representation exists: When is it possible to find a representation that uses fewer than \(2^s\) summands, in the extreme case a representation that is only linear in the \(h_i\)?
Answers to these questions are supplied in chapters 5 and 6. They include, among others, Schmüdgen’s theorem and a discussion of the moment problem, as well as results due to Jacobi and Prestel. – Chapter 7 shows how many results about representations with sums of squares can be extended to sums of \(2m\)-th powers, \(1\leq m\in\mathbb{N}\). Finally, chapter 8 deals with bounds for the representations discussed above, mainly concerning the degrees of the polynomials. The most desirable result would be a computable function – depending on the number of variables, on the number of polynomials defining the basic closed semialgebraic set, and on the degrees of the \(h_i\) and the polynomial to be represented – that gives a bound for the degrees of all polynomials that occur in a representation. It is shown that sometimes, but not always, such a function exists. If it does not exist, then the question is discussed what other parameters are needed to obtain a bound.

MSC:

13J30 Real algebra
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13J25 Ordered rings
11E10 Forms over real fields
12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
12J10 Valued fields
14P10 Semialgebraic sets and related spaces
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