Exponential lower bounds on spectrahedral representations of hyperbolicity cones

TL;DR

This paper establishes exponential lower bounds on the size of spectrahedral representations of hyperbolicity cones, indicating significant complexity in representing these cones with semidefinite programming.

Contribution

It proves that hyperbolicity cones of degree d in n variables require exponentially large spectrahedral representations, advancing understanding of the Generalized Lax Conjecture.

Findings

Hyperbolicity cones contain exponentially many distant cones.

Semidefinite representations must have exponential dimension.

Identification of a large subspace with elementary symmetric polynomials.

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 $(n/d)^{\Omega(d)}$ pairwise distant cones in a certain metric, and therefore that any semidefinite representation of such cones must have dimension at least $(n/d)^{\Omega(d)}$ (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.