Characteristic Polynomials of Symmetric Matrices over the Univariate Polynomial Ring

TL;DR

This paper proves that every real rooted bivariate polynomial can be represented as the characteristic polynomial of a symmetric matrix over R[x], and provides an algebraic proof of the Helton-Vinnikov Theorem related to hyperbolic forms.

Contribution

It establishes the converse of a known fact about characteristic polynomials and symmetric matrices, and offers a new algebraic proof of a longstanding theorem.

Findings

Every real rooted bivariate polynomial is a characteristic polynomial of a symmetric matrix over R[x]

Provides an algebraic proof of the Helton-Vinnikov Theorem

Solves the 60-year-old Lax conjecture for ternary hyperbolic forms

Abstract

Viewing a bivariate polynomial f in R[x,t] as a family of univariate polynomials in t parametrized by real numbers x, we call f real rooted if this family consists of monic polynomials with only real roots. If f is the characteristic polynomial of a symmetric matrix with entries in R[x], it is obviously real rooted. In this article the converse is established, namely that every real rooted bivariate polynomial is the characteristic polynomial of a symmetric matrix over the univariate real polynomial ring. As a byproduct we present a purely algebraic proof of the Helton-Vinnikov Theorem which solved the 60 year old Lax conjecture on the existence of definite determinantal representation of ternary hyperbolic forms.