Algebraic characterization of non-negativity of polynomials over polytopes

In this work we provide an algebraic method of recognizing polynomial ideals with structures that are conducive to the study of sum of squares (SOS) polynomials. Such ideals correspond to regions in space over which certificates of non-negativity may be efficiently obtained. As such, these regions are of great interest in the context of polynomial optimization. A geometric method of recognizing these regions was developed in 2011. Our contribution allows researchers to recognize this property based solely on the algebraic structure of the corresponding ideals. This allows researchers to study non-negativity without the need for expensive geometric computations.