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 2 y such that there are non-negative polynomials of degree 2 y 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 Bézoutians of hyperbolic polynomials.