Computing Linear Matrix Representations of Helton-Vinnikov Curves

Understand

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.