Vol. 303, No. 1, 2019

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Spectrahedral representations of plane hyperbolic curves

Mario Kummer, Simone Naldi and Daniel Plaumann

Vol. 303 (2019), No. 1, 243–263
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.

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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
Authors
Mario Kummer
Institut für Mathematik
Technische Universität Berlin
Berlin
Germany
Simone Naldi
XLIM
Université de Limoges
Limoges
France
Daniel Plaumann
Fakultät für Mathematik
Technische Universität Dortmund
Dortmund
Germany