Representations of non-negative polynomials having finitely many zeros

Marshall, Murray

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 ${\mathbf{R}}^n$ définie par les inégalités polynomiales $g_1\ge 0,...,g_s\ge 0$. 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 $f=t_0+t_1{g}_1+\cdots +t_s{g}_s$, 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 $g_1\ge 0,\cdots ,g_s\ge 0$. 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 $f=t_0+t_1{g}_1+\cdots +t_s{g}_s$, $t_i$ 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.

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