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arXiv · 1711.11497

Exponential lower bounds on spectrahedral representations of hyperbolicity cones

Prasad Raghavendra·Nick Ryder·Nikhil Srivastava·Benjamin Weitz
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Abstract

The Generalized Lax Conjecture asks whether every hyperbolicity cone is a section of a semidefinite cone of sufficiently high dimension. We prove that the space of hyperbolicity cones of hyperbolic polynomials of degree $d$ in $n$ variables contains $(n/d)^{Ω(d)}$ pairwise distant cones in a certain metric, and therefore that any semidefinite representation of such cones must have dimension at least $(n/d)^{Ω(d)}$ (even if a small approximation is allowed). The proof contains several ingredients of independent interest, including the identification of a large subspace in which the elementary symmetric polynomials lie in the relative interior of the set of hyperbolic polynomials, and quantitative versions of several basic facts about real rooted polynomials.

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Prasad Raghavendra, Nick Ryder, Nikhil Srivastava, Benjamin Weitz. 2018-01-12. Exponential lower bounds on spectrahedral representations of hyperbolicity cones. https://arxiv.org/abs/1711.11497

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