Testing hyperbolicity of real polynomials

TL;DR

This paper explores methods to determine if a polynomial is hyperbolic by translating hyperbolicity into nonnegativity conditions and testing via sum-of-squares relaxations, addressing a challenging problem in polynomial analysis.

Contribution

It introduces new approaches to test hyperbolicity of polynomials through sum-of-squares relaxations, linking hyperbolicity to nonnegativity conditions.

Findings

Hyperbolicity can be characterized by nonnegativity conditions.

Sum-of-squares relaxations provide a practical testing framework.

The methods facilitate hyperbolicity testing for complex polynomials.

Abstract

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating hyperbolicity into nonnegativity conditions, which can then be tested via sum-of-squares relaxations.