  We prove a generalization of the Hermitian version of the Helton-Vinnikov
determinantal representation of hyperbolic polynomials to the class of
semi-hyperbolic polynomials, a strictly larger class, as shown by an example.
We also prove that certain hyperbolic polynomials affine in two out of four
variables divide a determinantal polynomial. The proofs are based on work
related to polynomials with no zeros on the bidisk and tridisk.
