Stable polynomials and admissible numerators in product domains

Given a polynomial p with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials q with the property that the rational function $q/p$ is bounded near a boundary zero of p. We give a complete description of this ideal of numerators in the case where the zero set of p is smooth and satisfies a non-degeneracy condition. In three variables, we give a description of the ideal in terms of an integral closure when p has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.