  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.
