Symmetric Hyperbolic Polynomials

TL;DR

This paper characterizes symmetric hyperbolic polynomials of degrees 3 and 4, explores their properties, and connects hyperbolicity cones to spectrahedral representations, advancing understanding of these polynomials in algebra and optimization.

Contribution

It provides a complete characterization of degree 3 symmetric hyperbolic polynomials and a large class of degree 4, introducing new links to linear maps and spectrahedral cones.

Findings

Complete characterization of degree 3 symmetric hyperbolic polynomials

Identification of spectrahedral hyperbolicity cones for certain symmetric hyperbolic cubics

Connection between hyperbolicity testing and the degree principle for symmetric polynomials

Abstract

Hyperbolic polynomials have been of recent interest due to applications in a wide variety of fields. We seek to better understand these polynomials in the case when they are symmetric, i.e. invariant under all permutations of variables. We give a complete characterization of the set of symmetric hyperbolic polynomials of degree 3, and a large class of symmetric hyperbolic polynomials of degree 4. For a class of polynomials, which we call hook-shaped, we relate symmetric hyperbolic polynomials to a class of linear maps of univariate polynomials preserving hyperbolicity, and give evidence toward a beautiful characterization of all such hook-shaped symmetric hyperbolic polynomials. We show that hyperbolicity cones of a class of symmetric hyperbolic polynomials, including all symmetric hyperbolic cubics, are spectrahedral. Finally, we connect testing hyperbolicity of a symmetric polynomial to the degree principle for symmetric nonnegative polynomials.