Computing Linear Matrix Representations of Helton-Vinnikov Curves

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.

19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Basel

References in corpus (3)

• The Lax conjecture is true
• Quartic Curves and Their Bitangents
• On the Torelli problem and Jacobian Nullwerte in genus three

Cited by in corpus (8)

• Quartic Curves and Their Bitangents
• Determinantal representations of hyperbolic plane curves: An elementary approach
• Algebraic boundaries of Hilbert's SOS cones
• Minimal determinantal representations of bivariate polynomials
• Simple determinantal representations of up to quintic bivariate polynomials
• The numerical range of a periodic tridiagonal operator reduces to the numerical range of a finite matrix
• Computing symmetric determinantal representations
• Definite Determinantal Representations of Multivariate Polynomials