Nichtnegativstellensätze for definable functions in o-minimal structures

Abstract

This paper addresses to Nichtnegativstellensätze for definable functions in o-minimal structures on $(\mathbb{R},+,\cdot ).$ Namely, let $f,g_1,\dots ,g_l:{\mathbb{R}}^n\to \mathbb{R}$ be definable $C^p$-functions ($p\ge 2$) and assume that $f$ is non-negative on $S:=\{x\in {\mathbb{R}}^n|g_1(x)\ge 0,\dots ,g_l(x)\ge 0\}.$ Under some natural hypotheses on zeros of $f$ in $S,$ we show that $f$ is expressible in the form $f={\varphi }_0+{\sum }_{i=1}^l{\varphi }_i{g}_i,$ where each ${\varphi }_i$ is a sum of squares of definable $C^{p-2}$-functions. As a consequence, we derive global optimality conditions which generalize the Karush--Kuhn--Tucker optimality conditions for nonlinear optimization.