Determinantal representations of semi-hyperbolic polynomials

TL;DR

This paper extends the Hermitian determinantal representation from hyperbolic to semi-hyperbolic polynomials, broadening the class of polynomials with such representations and exploring divisibility properties related to hyperbolic polynomials.

Contribution

It generalizes the Helton-Vinnikov determinantal representation to semi-hyperbolic polynomials and investigates divisibility of certain hyperbolic polynomials by determinantal polynomials.

Findings

Generalization of determinantal representation to semi-hyperbolic polynomials

Existence of divisibility of hyperbolic polynomials affine in two variables

Connections to polynomials with no zeros on bidisk and tridisk

Abstract

We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.