Computing linear matrix representations of Helton-Vinnikov curves. (English) Zbl 1328.14093

Plaumann, Daniel; Sturmfels, Bernd; Vinzant, Cynthia

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].

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