Determinantal representations of hyperbolic plane curves: An elementary approach

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If a real symmetric matrix of linear forms is positive definite at some point, then its determinant is a hyperbolic hypersurface. In 2007, Helton and Vinnikov proved a converse in three variables, namely that every hyperbolic plane curve has a definite real symmetric determinantal representation. The goal of this paper is to give a more concrete proof of a slightly weaker statement. Here we show that every hyperbolic plane curve has a definite determinantal representation with Hermitian matrices. We do this by relating the definiteness of a matrix to the real topology of its minors and extending a construction of Dixon from 1902. Like Helton and Vinnikov's theorem, this implies that every hyperbolic region in the plane is defined by a linear matrix inequality.