Computing Linear Matrix Representations of Helton-Vinnikov Curves
Algebraic Geometry
2013-12-03 v2 Computational Geometry
Optimization and Control
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.
Keywords
Cite
@article{arxiv.1011.6057,
title = {Computing Linear Matrix Representations of Helton-Vinnikov Curves},
author = {Daniel Plaumann and Bernd Sturmfels and Cynthia Vinzant},
journal= {arXiv preprint arXiv:1011.6057},
year = {2013}
}
Comments
19 pages, 3 figures, minor revisions; Mathematical Methods in Systems, Optimization and Control, Birkhauser, Basel