Non-negative polynomials without hyperbolic certificates of non-negativity

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.