  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.
