Polynomials Positive on Unbounded Rectangles

Abstract

Given a semialgebraic set $K \subseteq \mathbb{R}^{N}$ determined by a finite set of polynomial inequalities {g₁ ≥ 0, ..., g _k ≥ 0} , we want to characterize a polynomial f which is positive (or non-negative) on K in terms of sums of squares and the polynomials g i used to describe K. Such a representation of f is an immediate witness to the positivity condition. Theorems about the existence of such representations also have various applications, notably in problems of optimizing polynomial functions on semialgebraic sets.