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 $q\mathrm{/}p$ 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. We also 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.