Determinantal Representations and B\'ezoutians

TL;DR

This paper proves that for every smooth hyperbolic polynomial, there exists another hyperbolic polynomial such that their product admits a definite determinantal representation, using sum-of-squares decompositions of Be9zoutians.

Contribution

It establishes a new connection between hyperbolic polynomials and determinantal representations via Be9zoutians, combining algebraic and geometric methods.

Findings

Existence of hyperbolic polynomials with definite determinantal representations

Use of sum-of-squares decompositions of Be9zoutians in the proof

Integration of commutative algebra and real algebraic geometry techniques

Abstract

We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by considering sum-of-squares decompositions of certain bilinear forms called B\'ezoutians. Besides commutative algebra, the proof relies on results from real algebraic geometry.