Remarks on the Lax conjecture for hyperbolic polynomials

A conjecture of Lax [P. Lax, Differential equations, difference equations and matrix theory, Commun. Pure Appl. Math. 11 (1958) 175–194] says that every hyperbolic polynomial in two space variables is the determinant of a symmetric hyperbolic matrix. The conjecture has recently been proved by Lewis–Parillo–Ramana, based on previous work of Dubrovin and Helton–Vinnikov. In this note we prove related results for polynomials in several space variables which have rotational symmetries.