Consider a compact subset of real -space defined by polynomial inequalities . For a polynomial non-negative on , natural sufficient conditions are given (in terms of first and second derivatives at the zeros of in ) for to have a presentation of the form , 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 une partie compacte de définie par les inégalités polynomiales . Pour un polynôme positif sur , des conditions suffisantes naturelles sont dégagées (en termes des dérivées premières et secondes en les zéros de dans ) pour que puisse se représenter sous la forme , où les 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.
@article{AFST_2006_6_15_3_599_0, author = {Marshall, Murray}, title = {Representations of non-negative polynomials having finitely many zeros}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {599--609}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 15}, number = {3}, year = {2006}, doi = {10.5802/afst.1131}, zbl = {1130.13015}, mrnumber = {2246416}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1131/} }
TY - JOUR AU - Marshall, Murray TI - Representations of non-negative polynomials having finitely many zeros JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2006 SP - 599 EP - 609 VL - 15 IS - 3 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1131/ DO - 10.5802/afst.1131 LA - en ID - AFST_2006_6_15_3_599_0 ER -
%0 Journal Article %A Marshall, Murray %T Representations of non-negative polynomials having finitely many zeros %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2006 %P 599-609 %V 15 %N 3 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1131/ %R 10.5802/afst.1131 %G en %F AFST_2006_6_15_3_599_0
Marshall, Murray. 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. http://archive.numdam.org/articles/10.5802/afst.1131/
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