Copositivity, discriminants and nonseparable signed supports

Abstract

We establish a connection between discriminants and copositivity of sparse Laurent polynomials. More generally, we consider signomials, which are defined on the positive orthant by linear combinations of monomials with real exponents, and their signed support, consisting of the support of the signomial and the sign of the coefficients. We provide a criterion to determine whether a given signomial is copositive, that is, only attains nonnegative values, which relies on finding the first intersection point in coefficient space of a path and a signed discriminant: the path contains the coefficient vector of the signomial and preserves its signs, and the signed discriminant encodes the singular signomials with the given signed support. If the signed support satisfies a combinatorial condition termed nonseparability, we additionally show that this intersection consists of one point, and that tracking one path with homotopy continuation suffices to decide upon copositivity. Building on these results, we show that copositive polynomials with nonseparable signed support can always be decomposed into a sum of nonnegative circuit polynomials, generalising thereby previously known supports having this property.