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Sum of squares and the Łojasiewicz exponent at infinity

Tom 112 / 2014

Annales Polonici Mathematici 112 (2014), 223-237 MSC: 14R99, 11E25, 14P05, 32S70. DOI: 10.4064/ap112-3-2

Streszczenie

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.

Autorzy

• Krzysztof KurdykaLaboratoire de Mathématiques (LAMA)
Université de Savoie
UMR-5127 de CNRS
73-376 Le Bourget-du-Lac Cedex, France
e-mail
• Beata Osińska-UlrychFaculty of Mathematics
and Computer Science
University of Łódź
90-238 Łódź, Poland
e-mail
• Grzegorz SkalskiFaculty of Mathematics
and Computer Science
University of Łódź
90-238 Łódź, Poland
e-mail
• Stanisław SpodziejaFaculty of Mathematics
and Computer Science
University of Łódź
90-238 Łódź, Poland
e-mail

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