S.o.s. approximation of polynomials nonnegative on a real algebraic set

Abstract

With every real polynomial $f$, we associate a family $\{f_{\epsilon r}\}_{\epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $\epsilon>0,r\in N$, and such that $\Vert f-f_{\epsilon r}\Vert_1\to 0$ as $\epsilon\to 0$. Let $V\subset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $\epsilon>0$, there exist nonnegative scalars $\{\lambda_j(\epsilon)\}_{j\in J}$ such that, for all $r$ sufficiently large, $$f_{\epsilon r}+\sum_{j\in J} \lambda_j(\epsilon) g_j^2,\quad is a sum of squares.$$ This representation is an obvious certificate of nonnegativity of $f_{\epsilon r}$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with {\it no} assumption on $V$. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a semi-algebraic set $K$, and again, with {\it no} assumption on $V$ or $K$.