Stable polynomials and bounded rational functions in the unit ball

TL;DR

This paper investigates polynomials without zeros in the unit ball and characterizes when rational functions are bounded there, providing local descriptions, partial higher-dimensional results, and various applications.

Contribution

It offers a complete local description of boundary-zero polynomials in two variables and partial results in higher dimensions, advancing understanding of bounded rational functions.

Findings

Complete local description of boundary-zero polynomials in two variables

Partial characterization of boundary zeros in higher dimensions

Applications to boundedness and example construction of rational functions

Abstract

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.