Computing Linear Matrix Representations of Helton-Vinnikov Curves

TL;DR

This paper explores three methods—algebraic, geometric, and analytic—for computing symmetric linear matrix representations of Helton-Vinnikov curves, with experimental comparisons for low-degree cases.

Contribution

It provides a comprehensive analysis and comparison of algebraic, geometric, and analytic approaches to explicitly compute matrix representations of Helton-Vinnikov curves.

Findings

All three approaches are feasible for low-degree curves.

Experimental results compare the efficiency and accuracy of each method.

The paper offers insights into the advantages and limitations of each approach.

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.