Determinantal representations of hyperbolic curves via polynomial homotopy continuation

TL;DR

This paper introduces a numerical method using polynomial homotopy continuation to compute definite symmetric determinantal representations of hyperbolic curves in the real projective plane, leveraging the Helton-Vinnikov Theorem.

Contribution

It presents a novel numerical approach to find determinantal representations of hyperbolic curves via polynomial homotopy continuation, expanding computational tools in algebraic geometry.

Findings

Successfully computes determinantal representations for hyperbolic curves

Demonstrates the effectiveness of homotopy continuation in this geometric context

Provides a practical method for applications in optimization and real algebraic geometry

Abstract

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.