Spectrahedrality of hyperbolicity cones of multivariate matching polynomials

TL;DR

This paper proves the generalized Lax conjecture for certain multivariate polynomials, showing their hyperbolicity cones are spectrahedral, and extends these results to related polynomials with applications to symmetric polynomials.

Contribution

It establishes the spectrahedrality of hyperbolicity cones for multivariate matching and independence polynomials, advancing the understanding of the generalized Lax conjecture.

Findings

Proves the generalized Lax conjecture for multivariate matching polynomials.

Extends results to multivariate independence polynomials for simplicial graphs.

Provides a new proof for the conjecture on elementary symmetric polynomials.

Abstract

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Br\"and\'en). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.