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 polynomials must have dimension at least $(n/d)^{\Omega(d)}$. 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. 查看全文>>