Type of Publication: | Journal article |
Author: | Plaumann, Daniel; Vinzant, Cynthia |
Year of publication: | 2013 |
Published in: | Journal of Symbolic Computation ; 57 (2013). - pp. 48-60. - ISSN 0747-7171. - eISSN 1095-855X |
DOI (citable link): | https://dx.doi.org/10.1016/j.jsc.2013.05.004 |
Summary: |
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.
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Subject (DDC): | 510 Mathematics |
Keywords: | Hyperbolic polynomials, determinantal representations, interlacing, Hermitian matrices of linear forms |
Bibliography of Konstanz: | Yes |
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PLAUMANN, Daniel, Cynthia VINZANT, 2013. Determinantal representations of hyperbolic plane curves : an elementary approach. In: Journal of Symbolic Computation. 57, pp. 48-60. ISSN 0747-7171. eISSN 1095-855X. Available under: doi: 10.1016/j.jsc.2013.05.004
@article{Plaumann2013Deter-26402, title={Determinantal representations of hyperbolic plane curves : an elementary approach}, year={2013}, doi={10.1016/j.jsc.2013.05.004}, volume={57}, issn={0747-7171}, journal={Journal of Symbolic Computation}, pages={48--60}, author={Plaumann, Daniel and Vinzant, Cynthia} }
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