Copositivity, discriminants and nonseparable signed supports

TL;DR

This paper links copositivity of sparse Laurent polynomials to discriminants, providing a criterion for copositivity based on discriminant intersection points and a homotopy method for nonseparable supports.

Contribution

It introduces a discriminant-based criterion for copositivity and extends the decomposition of copositive polynomials into nonnegative circuit polynomials.

Findings

Discriminant intersection points determine copositivity.

Homotopy continuation suffices for nonseparable supports.

Decomposition into nonnegative circuit polynomials is generalized.

Abstract

In this work we establish a connection between copositivity, that is, nonnegativity on the positive orthant, of sparse real Laurent polynomials and discriminants. Specifically, we consider Laurent polynomials in the positive orthant with fixed support and fixed coefficient signs. We provide a criterion to decide whether a given polynomial is copositive that is based in determining the intersection points of the signed discriminant and a path going through the coefficients of the polynomial. If the signed support satisfies a combinatorial condition termed nonseparability, we show additionally that this intersection consists of one point, and that tracking one path in homotopy continuation methods suffices to decide upon copositivity. Building on these results, we show that any copositive polynomial with nonseparable signed support can be decomposed into a sum of nonnegative circuit polynomials, generalising thereby previously known supports having this property.