Non-negative polynomials without hyperbolic certificates of non-negativity

TL;DR

This paper explores the relationship between non-negative multivariate homogeneous polynomials and hyperwrons, revealing the existence of non-negative polynomials that are not hyperwrons and providing explicit examples and conditions.

Contribution

It introduces conditions under which non-negative polynomials are not hyperwrons and provides explicit examples, advancing understanding of polynomial non-negativity certificates.

Findings

Existence of non-negative polynomials that are not hyperwrons under certain conditions

Explicit example of a non-negative quartic form not expressible as a sum of hyperwrons

Partial extension of results to hyperzouts involving Bézoutians

Abstract

In this paper we study the relationship between the set of all non-negative multivariate homogeneous polynomials and those, which we call hyperwrons, whose non-negativity can be deduced from an identity involving the Wronskians of hyperbolic polynomials. We give a sufficient condition on positive integers $m$ and $2y$ such that there are non-negative polynomials of degree $2y$ in $m$ variables that are not hyperwrons. Furthermore, we give an explicit example of a non-negative quartic form that is not a sum of hyperwrons. We partially extend our results to hyperzouts, which are polynomials whose non-negativity can be deduced from an identity involving the B\'ezoutians of hyperbolic polynomials.