Stable polynomials and admissible numerators in product domains

Given a polynomial (Formula presented.) with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials (Formula presented.) with the property that the rational function (Formula presented.) is bounded near a boundary zero of (Formula presented.). We give a complete description of this ideal of numerators in the case where the zero set of (Formula presented.) is smooth and satisfies a nondegeneracy condition. We also give a description of the ideal in terms of an integral closure when (Formula presented.) 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.