Publikation: Determinantal representations of hyperbolic curves via polynomial homotopy continuation
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2017
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Leykin, Anton
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Mathematics of Computation. 2017, 86(308), pp. 2877-2888. ISSN 0025-5718. eISSN 1088-6842. Available under: doi: 10.1090/mcom/3194
Zusammenfassung
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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LEYKIN, Anton, Daniel PLAUMANN, 2017. Determinantal representations of hyperbolic curves via polynomial homotopy continuation. In: Mathematics of Computation. 2017, 86(308), pp. 2877-2888. ISSN 0025-5718. eISSN 1088-6842. Available under: doi: 10.1090/mcom/3194BibTex
@article{Leykin2017Deter-39627,
year={2017},
doi={10.1090/mcom/3194},
title={Determinantal representations of hyperbolic curves via polynomial homotopy continuation},
number={308},
volume={86},
issn={0025-5718},
journal={Mathematics of Computation},
pages={2877--2888},
author={Leykin, Anton and Plaumann, Daniel}
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<dcterms:abstract xml:lang="eng">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.</dcterms:abstract>
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