Computing Hermitian determinantal representations of hyperbolic curves

TL;DR

This paper introduces a computationally efficient algorithm for finding Hermitian determinantal representations of hyperbolic curves, providing an algebraic certificate for hyperbolicity with practical numerical implementation.

Contribution

It presents a new algorithm that simplifies the computation of determinantal representations, reducing the problem to linear algebra and enhancing numerical feasibility.

Findings

Algorithm effectively computes Hermitian determinantal representations

Reduces computational complexity compared to previous methods

Demonstrates successful numerical implementation

Abstract

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the polynomial and their existence has been proved in several different ways. However, the resulting algorithms for computing determinantal representations are computationally intensive. In this note, we present an algorithm that reduces a large part of the problem to linear algebra and discuss its numerical implementation.