Exponential lower bounds on spectrahedral representations of hyperbolicity cones

11/30/2017

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by   Prasad Raghavendra, et al.

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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 polynomials must have dimension at least (n/d)^Ω(d). 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.