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Computing Linear Matrix Representations of Helton-Vinnikov Curves

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Mathematical Methods in Systems, Optimization, and Control

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 222))

Abstract

Helton and Vinnikov showed that every rigidly convex curve in the real plane bounds a spectrahedron. This leads to the computational problem of explicitly producing a symmetric (positive definite) linear determinantal representation for a given curve. We study three approaches to this problem: an algebraic approach via solving polynomial equations, a geometric approach via contact curves, and an analytic approach via theta functions. These are explained, compared, and tested experimentally for low degree instances.

Mathematics Subject Classification. Primary: 14Q05; secondary: 14K25.

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik und Statistik, Universität Konstanz, D-78457, Konstanz, Germany

    Daniel Plaumann

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Bernd Sturmfels

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Cynthia Vinzant

Authors
  1. Daniel Plaumann
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  2. Bernd Sturmfels
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  3. Cynthia Vinzant
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Corresponding author

Correspondence to Daniel Plaumann .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Harry Dym

  2. , Department of Mechanical and, University of California San Diego, Gilman Drive 9500, La Jolla, 92093-0411, California, USA

    Mauricio C. de Oliveira

  3. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

Additional information

Dedicated to Bill Helton on the occasion of his 65th birthday

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Plaumann, D., Sturmfels, B., Vinzant, C. (2012). Computing Linear Matrix Representations of Helton-Vinnikov Curves. In: Dym, H., de Oliveira, M., Putinar, M. (eds) Mathematical Methods in Systems, Optimization, and Control. Operator Theory: Advances and Applications, vol 222. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0411-0_19

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