Definite Determinantal Representations via Orthostochastic Matrices

TL;DR

This paper introduces a new criterion based on orthostochastic matrices for determining the existence of definite determinantal representations of bivariate polynomials, and develops methods for their computation.

Contribution

It provides a necessary and sufficient condition for determinantal representations using orthostochastic matrices and proposes a computational relaxation approach.

Findings

Characterizes when a bivariate polynomial admits a determinantal representation.

Develops a method to compute symmetric/Hermitian determinantal representations.

Introduces a convex combination approach for the coefficients of the polynomial.

Abstract

Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a determinantal representation. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree $d$. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points. We also characterize the range set of vector coefficients of a certain type of determinantal bivariate polynomials.