The Generalized Lax Conjecture for hyperbolicity cones

A homogeneous polynomial $h\in\mathbb{R}[x_1,\ldots,x_n]$ is hyperbolic with respect to $e\in\mathbb{R}^n$ if $h(e)>0$ and, for every $a\in\mathbb{R}^n$, the polynomial $h(te-a)$ has only real roots. Its hyperbolicity cone is

$$\operatorname*{C}(h,e)=\{a\in\mathbb{R}^n:\text{ all zeros of }h(te-a)\text{ are nonnegative}\}.$$

A cone is spectrahedral if it is defined by a linear matrix inequality, equivalently as the positive-semidefinite solution set of a linear matrix pencil. Generalized Lax Conjecture. Every hyperbolicity cone is spectrahedral. This is a major open problem in the theory of hyperbolic polynomials and convex optimization. The paper proves spectrahedrality for derivative relaxations of the positive semidefinite cone under a representation-theoretic assumption, but the general conjecture remains open.