Definite determinantal representations via orthostochastic matrices

Abstract

Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate polynomials are definite determinantals. Indeed, in general, it is a difficult problem to decide whether a given polynomial is definite determinantal, and if it is, it is of paramount interest to determine a definite determinantal representation of that polynomial. 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.