Determinantal representations of hyperbolic curves via polynomial homotopy continuation

Leykin, Anton; Plaumann, Daniel

doi:10.1090/mcom/3194

Determinantal representations of hyperbolic curves via polynomial homotopy continuation
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by Anton Leykin and Daniel Plaumann PDF

Math. Comp. 86 (2017), 2877-2888 Request permission

Abstract:

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.

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