Non-negative polynomials without hyperbolic certificates of non-negativity

2025/08/06 by Ng, H. L. Brian, Saunderson, James
#Algebraic Geometry (math.AG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2508.04027

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 Bézoutians of hyperbolic polynomials.

Citations

• On Sum of Squares Representation of Convex Forms and Generalized\n Cauchy-Schwarz Inequalities
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• Complete family of separability criteria

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