Determinantal representations of hyperbolic curves via polynomial homotopy continuation
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- by Anton Leykin and Daniel Plaumann PDF
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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.References
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Additional Information
- Anton Leykin
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia
- MR Author ID: 687160
- ORCID: 0000-0002-9216-3514
- 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
- MR Author ID: 894950
- Email: Daniel.Plaumann@math.tu-dortmund.de
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 2877-2888
- MSC (2010): Primary 14P99, 14Q05, 14Q99; Secondary 65F40, 90C22
- DOI: https://doi.org/10.1090/mcom/3194
- MathSciNet review: 3667028