Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets

Abstract

We prove a criterion for an element of a commutative ring A to be contained in an archimedean semiring T A. It can be used to investigate the question whether nonnegativity of a polynomial on a compact semialgebraic set can be certified in a certain way. In case of (strict) positivity instead of nonnegativity, our criterion simplifies to classical results of Stone, Kadison, Krivine, Handelman, Schmudgen et al. As an application of our result, we give a new proof of the following result of Handelman: If an odd power of a real polynomial in several variables has only nonnegative coecients, then so do all suciently high powers.