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Representations of non-negative polynomials having finitely many zeros
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 599-609.

Soit K une partie compacte de R n définie par les inégalités polynomiales g 1 0,...,g s 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 ++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.

Consider a compact subset K of real n-space defined by polynomial inequalities g 1 0,,g s 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 ++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.

DOI : 10.5802/afst.1131
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, Série 6, Tome 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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