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Mathematics of Computation

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Determinantal representations of hyperbolic curves via polynomial homotopy continuation


Authors: Anton Leykin and Daniel Plaumann
Journal: Math. Comp. 86 (2017), 2877-2888
MSC (2010): Primary 14P99, 14Q05, 14Q99; Secondary 65F40, 90C22
DOI: https://doi.org/10.1090/mcom/3194
Published electronically: February 16, 2017
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Abstract: A smooth curve of degree $ d$ in the real projective plane is hyperbolic if its ovals are maximally nested, i.e., its real points contain $ \lfloor \frac d2\rfloor $ nested ovals. 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.


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Additional Information

Anton Leykin
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia
Email: leykin@math.gatech.edu

Daniel Plaumann
Affiliation: Fachbereich Mathematik und Statistik, Universität Konstanz, Germany
Address at time of publication: Technische Universität Dortmund, Fakultät für Mathematik, 44227 Dortmund, Germany
Email: Daniel.Plaumann@math.tu-dortmund.de

DOI: https://doi.org/10.1090/mcom/3194
Received by editor(s): November 25, 2014
Received by editor(s) in revised form: June 26, 2016
Published electronically: February 16, 2017
Additional Notes: The first author was supported by NSF grant DMS-1151297
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society