Determinantal representations of plane curves via homotopy continuation

Determinantal representations of plane curves via polynomial homotopy continuation

Anton Leykin and Daniel Plaumann

A smooth curve of degree $d$ in the real projective plane is hyperbolic if its ovals are maximally nested, i.e. its real points contain $\lfloor\frac d2\rfloor$ nested ovals. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

(Figure on the right: a quartic following the N-path.)

Paper: arXiv:1212.3506

Scripts and examples:

Main file used by the others: HyperbolicPolynomialHomotopy.m2
Examples of computation: showcase.m2
A batch of examples used for experimentation and timings: TIMINGS.tar
Proof of positivity of the discriminant on the N-path in case $d=2$: Nuij-d2.mathematica

Needed software:

Macaulay2 version 1.9+
Mathematica version 8+

(Last updated: June 2016)