Lax's conjecture on determinantal representations of ternary hyperbolic polynomials

Let $p$ be a polynomial on $\mathbb{R}^3$ that is hyperbolic of degree $d$ with respect to $e=(1,0,0)$ and satisfies $p(e)=1$. Let $I$ denote the $d\times d$ identity matrix and let $\mathbb{S}^d$ denote the space of real symmetric $d\times d$ matrices. Lax's conjecture. There exist matrices $B,C\in\mathbb{S}^d$ such that

$$p(x,y,z)=\det(xI+yB+zC).$$

The conjecture asserts that every normalized ternary hyperbolic polynomial has a definite symmetric determinantal representation, equivalently that its hyperbolicity cone is a semidefinite slice. The paper proves this conjecture, so the statement is now a theorem.