Plaumann, Daniel; Sturmfels, Bernd; Vinzant, Cynthia Computing linear matrix representations of Helton-Vinnikov curves. (English) Zbl 1328.14093 Dym, Harry (ed.) et al., Mathematical methods in systems, optimization, and control. Festschrift in honor of J. William Helton. Basel: Birkhäuser (ISBN 978-3-0348-0410-3/hbk; 978-3-0348-0411-0/ebook). Operator Theory: Advances and Applications 222, 259-277 (2012). Summary: J. W. Helton and V. Vinnikov [Commun. Pure Appl. Math. 60, No. 5, 654–674 (2007; Zbl 1116.15016)] 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.For the entire collection see [Zbl 1250.00008]. Cited in 23 Documents MSC: 14Q05 Computational aspects of algebraic curves 14K25 Theta functions and abelian varieties 14P05 Real algebraic sets 14M12 Determinantal varieties Keywords:plane curves; symmetric determinantal representations; spectrahedra; linear matrix inequalities; hyperbolic polynomials; theta functions Software:Bertini; SINGULAR PDF BibTeX XML Cite \textit{D. Plaumann} et al., Oper. Theory: Adv. Appl. 222, 259--277 (2012; Zbl 1328.14093) Full Text: DOI arXiv