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arXiv · 2308.09653

Symmetric Hyperbolic Polynomials

Grigoriy Blekherman·Julia Lindberg·Kevin Shu
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Abstract

Hyperbolic polynomials have been of recent interest due to applications in a wide variety of fields. We seek to better understand these polynomials in the case when they are symmetric, i.e. invariant under all permutations of variables. We give a complete characterization of the set of symmetric hyperbolic polynomials of degree 3, and a large class of symmetric hyperbolic polynomials of degree 4. For a class of polynomials, which we call hook-shaped, we relate symmetric hyperbolic polynomials to a class of linear maps of univariate polynomials preserving hyperbolicity, and give evidence toward a beautiful characterization of all such hook-shaped symmetric hyperbolic polynomials. We show that hyperbolicity cones of a class of symmetric hyperbolic polynomials, including all symmetric hyperbolic cubics, are spectrahedral. Finally, we connect testing hyperbolicity of a symmetric polynomial to the degree principle for symmetric nonnegative polynomials.

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Grigoriy Blekherman, Julia Lindberg, Kevin Shu. 2023-08-18. Symmetric Hyperbolic Polynomials. https://arxiv.org/abs/2308.09653

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