Spectrahedral representations of plane hyperbolic curves

Mario Kummer, Simone Naldi and Daniel Plaumann

Vol. 303 (2019), No. 1, 243–263

DOI: 10.2140/pjm.2019.303.243

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.

Keywords

real algebraic curves, determinantal representations, spectrahedra, linear matrix inequalities

Mathematical Subject Classification 2010

Primary: 14H50, 14P99, 52A10

Milestones

Received: 28 August 2018

Revised: 30 May 2019

Accepted: 8 June 2019

Published: 21 December 2019