Determinantal representations of hyperbolic plane curves : an elementary approach

Plaumann, Daniel; Vinzant, Cynthia

Determinantal representations of hyperbolic plane curves : an elementary approach

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Datum

2013

Publikationstyp

Zeitschriftenartikel

Publikationsstatus

Published

Erschienen in

Journal of Symbolic Computation. 2013, 57, pp. 48-60. ISSN 0747-7171. eISSN 1095-855X. Available under: doi: 10.1016/j.jsc.2013.05.004

Zusammenfassung

In 2007, Helton and Vinnikov proved that every hyperbolic plane curve has a definite real symmetric determinantal representation. By allowing for Hermitian matrices instead, we are able to give a new proof that relies only on the basic intersection theory of plane curves. We show that a matrix of linear forms is definite if and only if its co-maximal minors interlace its determinant and extend a classical construction of determinantal representations of Dixon from 1902. Like the Helton-Vinnikov theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Hyperbolic polynomials, determinantal representations, interlacing, Hermitian matrices of linear forms

Zitieren

ISO 690

PLAUMANN, Daniel, Cynthia VINZANT, 2013. Determinantal representations of hyperbolic plane curves : an elementary approach. In: Journal of Symbolic Computation. 2013, 57, pp. 48-60. ISSN 0747-7171. eISSN 1095-855X. Available under: doi: 10.1016/j.jsc.2013.05.004

BibTex

@article{Plaumann2013Deter-26402,
  year={2013},
  doi={10.1016/j.jsc.2013.05.004},
  title={Determinantal representations of hyperbolic plane curves : an elementary approach},
  volume={57},
  issn={0747-7171},
  journal={Journal of Symbolic Computation},
  pages={48--60},
  author={Plaumann, Daniel and Vinzant, Cynthia}
}

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