Stable and real-zero polynomials in two variables

For every bivariate polynomial p ( z 1 , z 2 ) of bidegree ( n 1 , n 2 ) , with p ( 0 , 0 ) = 1 , which has no zeros in the open unit bidisk, we construct a determinantal representation of the form p ( z 1 , z 2 ) = det ( I - K Z ) , where Z is an ( n 1 + n 2 ) × ( n 1 + n 2 ) diagonal matrix with coordinate variables z 1 , z 2 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 p ( x 1 , x 2 ) , with p ( 0 , 0 ) = 1 , we provide a construction to build a representation of the form p ( x 1 , x 2 ) = det ( I + x 1 A 1 + x 2 A 2 ) , where A 1 and A 2 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).