Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka¹ ; Beata Osińska-Ulrych² ; Grzegorz Skalski² ; Stanisław Spodzieja²

¹ Laboratoire de Mathématiques (LAMA) Université de Savoie UMR-5127 de CNRS 73-376 Le Bourget-du-Lac Cedex, France
² Faculty of Mathematics and Computer Science University of Łódź 90-238 Łódź, Poland

Annales Polonici Mathematici, Tome 112 (2014) no. 3, pp. 223-237

Résumé

Let $V\subset \mathbf {\mathbb {R}}^n$, $n\ge 2$, be an unbounded algebraic set defined by a system of polynomial equations $h_1(x)=\cdots =h_r(x)=0$ and let $f:\mathbf {\mathbb {R}}^n\to \mathbf {\mathbb {R}}$ be a polynomial. It is known that if $f$ is positive on $V$ then $f|_V$ extends to a positive polynomial on the ambient space $\mathbf {\mathbb {R}}^n$, provided $V$ is a variety. We give a constructive proof of this fact for an arbitrary algebraic set $V$. Precisely, if $f$ is positive on $V$ then there exists a polynomial $h(x)=\sum_{i=1}^r h_i^2(x)\sigma _i(x)$, where $\sigma _i$ are sums of squares of polynomials of degree at most $p$, such that $f(x)+h(x)>0$ for $x\in \mathbf {\mathbb {R}}^n$. We give an estimate for $p$ in terms of: the degree of $f$, the degrees of $h_i$ and the Łojasiewicz exponent at infinity of $f|_V$. We prove a version of the above result for polynomials positive on semialgebraic sets. We also obtain a nonnegative extension of some odd power of $f$ which is nonnegative on an irreducible algebraic set.

Détail

DOI : 10.4064/ap112-3-2

Mots-clés : subset mathbf mathbb unbounded algebraic set defined system polynomial equations cdots mathbf mathbb mathbf mathbb polynomial known positive extends positive polynomial ambient space mathbf mathbb provided variety constructive proof arbitrary algebraic set precisely positive there exists polynomial sum sigma where sigma sums squares polynomials degree nbsp mathbf mathbb estimate terms degree degrees ojasiewicz exponent infinity prove version above result polynomials positive semialgebraic sets obtain nonnegative extension odd power which nonnegative irreducible algebraic set