A real Nullstellensatz and Positivstellensatz for the semipolynomials over an ordered field
type: Item
instance of: scholarly article
title: A real Nullstellensatz and Positivstellensatz for the semipolynomials over an ordered field
zbMATH Open document ID: 0826.12004
DOI: 10.1016/0022-4049(93)90128-G
author: Henri Lombardi
author: Laureano Gonzalez-Vega
published in: Journal of Pure and Applied Algebra
publication date: 1995-11-28T00:00:00Z
review text: Given a totally ordered field \(K\) with real closure \(R\), a variation of Hilbert's original 17-th problem asks whether given a polynomial \(P\in K[ X]= K[X_1, \dots, X_n]\) which takes nonnegative values at each point of \(\mathbb{R}^n\) (i.e., \(P\) is positive semidefinite on \(\mathbb{R}^n\)) is representable in the form \(P= \sum p_i r_i^2\) with \(0\leq p_i\in K\), \(r_i\in K(X)\) rational functions. After this problem had been solved the focus of further investigations shifted to the question of how the \(p_i\) and \(r_i\) depend on the coefficients of \(P\). The most far reaching solution was presented by \textit{C. N. Delzell} [J. Reine Angew. Math. 440, 157-173 (1993; Zbl 0774.12003)] who proved that the \(p_i\) as well as the coefficients of the polynomials in the numerators and the denominators of the \(r_i\) can be chosen to be sup-inf- polynomially definable continuous functions, i.e., certain piecewise polynomial functions. The present paper contains another proof of this result as well as of a related Positivstellensatz. The method of proof is constructive.
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 14P99
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 500592
zbMATH Keywords: real Nullstellensatz
zbMATH Keywords: real closed field
zbMATH Keywords: positive semidefinite polynomial
zbMATH Keywords: sums of squares
zbMATH Keywords: totally ordered field
zbMATH Keywords: sup-inf-polynomially definable continuous functions
zbMATH Keywords: piecewise polynomial functions
zbMATH Keywords: Positivstellensatz
reviewed by: Niels Schwartz
MaRDI profile type: Publication
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full work available at URL: 0022-4049(93)90128-g
OpenAlex ID: W2011404597
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description: scientific article; zbMATH DE number 500592