Determinantal Representations and the Hermite Matrix

TL;DR

This paper explores the relationship between determinantal representations of real polynomials and sums of squares decompositions of the Hermite matrix, establishing conditions for their existence and construction.

Contribution

It provides new insights into when and how definite determinantal representations can be constructed from Hermite matrices, including cases involving denominators.

Findings

If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares.

A determinantal representation can sometimes be constructed from a sums-of-squares decomposition of the Hermite matrix.

Definite determinantal representations always exist if denominators are allowed.

Abstract

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Hermite matrix. If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares. Conversely, we show how a determinantal representation can sometimes be constructed from a sums-of-squares decomposition of the Hermite matrix. We finally show that definite determinantal representations always exist, if one allows for denominators.