Stable and real-zero polynomials in two variables

For every bivariate polynomial (Formula Presented.) of bidegree (Formula Presented.), which has no zeros in the open unit bidisk, we construct a determinantal representation of the form (Formula Presented.) diagonal matrix with coordinate variables (Formula Presented.) on the diagonal and K is a contraction. We show that K may be chosen to be unitary if and only if p is a (unimodular) constant multiple of its reverse. Furthermore, for every bivariate real-zero polynomial (Formula Presented.), we provide a construction to build a representation of the form (Formula Presented.),where (Formula Presented.) are Hermitian matrices of size equal to the degree of p. A key component of both constructions is a stable factorization of a positive semidefinite matrix-valued polynomial in one variable, either on the circle (trigonometric polynomial) or on the real line (algebraic polynomial).