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

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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real algebraic curves, determinantal representations, spectrahedra, linear matrix inequalities
Mathematical Subject Classification 2010
Primary: 14H50, 14P99, 52A10
Received: 28 August 2018
Revised: 30 May 2019
Accepted: 8 June 2019
Published: 21 December 2019
Mario Kummer
Institut für Mathematik
Technische Universität Berlin
Simone Naldi
Université de Limoges
Daniel Plaumann
Fakultät für Mathematik
Technische Universität Dortmund