Spectrahedral representations of plane hyperbolic curves

TL;DR

This paper introduces a novel method for constructing spectrahedral representations of hyperbolic curves in the real projective plane, with implications for rational matrices if the curve is smooth and defined over the rationals.

Contribution

It generalizes Dixon's classical construction by leveraging properties of real hyperbolic curves to produce spectrahedral representations.

Findings

Spectrahedral representations can be constructed for hyperbolic curves.

Rational matrices are possible if the curve is smooth and defined over rationals.

The method extends classical determinantal representations to a broader class of curves.

Abstract

We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show that if the curve is smooth and defined over the rational numbers, then there is a spectrahedral representation with rational matrices. This generalizes a classical construction for determinantal representations of plane curves due to Dixon and relies on the special properties of real hyperbolic curves that interlace the given curve.