Mate Telek: SONC exactness and the positive A-discriminant

Abstract

A classical approach to certifying the non-negativity of a real polynomial is to express it as a sum of squares. The cone of polynomials that can be represented as a sum of squares is, in general, strictly contained within the cone of non-negative polynomials. The cases when these two cones coincide were characterized by Hilbert.

An alternative approach for certifying non-negativity is to express the polynomial as a Sum of Non-negative Circuits (SONC). In this talk, I will discuss necessary and sufficient combinatorial conditions under which the cone of SONC polynomials coincides with the cone of non-negative polynomials. Our approach builds on extending Viro’s patchworking to singular hypersurfaces and using properties of the positive A-discriminant. This talk is based on joint work with Timo de Wolff.