Generalized Lax conjecture for real-zero polynomials

Prove that every real-zero polynomial p with p(0)=1 admits a real-zero cofactor q and a symmetric determinantal representation pq=det(I_d+x_1A_1+\cdots+x_nA_n) such that the rigidly convex set of p is contained in the rigidly convex set of q.

Background

The dissertation studies rigidly convex sets associated with real-zero polynomials and seeks spectrahedral representations or relaxations of these sets. In two variables, the Helton–Vinnikov theorem provides determinantal representations, but the analogous statement in higher dimensions requires multiplication by a real-zero cofactor q.

The conjecture requires both a determinantal representation of pq and the containment of the original rigidly convex set within the cofactor’s rigidly convex set. The containment condition is essential because it preserves control over the relevant convex set. The text notes that the conjecture is known only in special cases and remains unresolved in general.