Representations of non-negative polynomials having finitely many zeros

Murray Marshall

doi:10.5802/afst.1131

Murray Marshall ¹

¹ Department of Computer Science, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 599-609.

Abstract
Abstract

Consider a compact subset $K$ of real $n$ -space defined by polynomial inequalities $g_{1} \geq 0, \dots, g_{s} \geq 0$ . For a polynomial $f$ non-negative on $K$ , natural sufficient conditions are given (in terms of first and second derivatives at the zeros of $f$ in $K$ ) for $f$ to have a presentation of the form $f = t_{0} + t_{1} g_{1} + \dots + t_{s} g_{s}$ , $t_{i}$ a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory and valuation theory. We also explain how the basic lemma of Kuhlmann, Marshall and Schwartz in [3] can be used to simplify the proof of Scheiderer’s main theorem, and compare the two approaches.

Soit $K$ une partie compacte de $R^{n}$ définie par les inégalités polynomiales $g_{1} \geq 0, ..., g_{s} \geq 0$ . Pour un polynôme positif $f$ sur $K$ , des conditions suffisantes naturelles sont dégagées (en termes des dérivées premières et secondes en les zéros de $f$ dans $K$ ) pour que $f$ puisse se représenter sous la forme $f = t_{0} + t_{1} g_{1} + \dots + t_{s} g_{s}$ , où les $t_{i}$ sont des sommes de carrés de polynômes. Les conditions sont bien plus générales que celles mises en évidence par Scheiderer dans [11, Cor. 2.6]. La démonstration utilise le théorème principal de Scheiderer [11] ainsi que des arguments de la théorie des formes quadratiques et de celle de la valuation. L’article explique également comment le lemme fondamental de Kuhlmann, Marshall et Schwartz [3] peut être mis à profit pour simplifier le théorème principal de Scheiderer, et compare les deux approches.

EuDML MR Zbl

DOI: 10.5802/afst.1131

Author's affiliations:

Murray Marshall ¹

¹ Department of Computer Science, University of Saskatchewan, Saskatoon, SK Canada, S7N 5E6

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Murray Marshall. Representations of non-negative polynomials having finitely many zeros. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 599-609. doi : 10.5802/afst.1131. https://afst.centre-mersenne.org/articles/10.5802/afst.1131/

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