Sum of squares and the Łojasiewicz exponent at infinity
Volume 112 / 2014
Abstract
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.
