Stable polynomials and bounded rational functions in the unit ball

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near a boundary zero. In higher dimensions, we give a partial characterization of a simple boundary zero. Several applications are given including boundedness of rational functions with boundary singularities and constructions of examples with prescribed local properties.