Stable polynomials and admissible numerators in product domains

Publication Date

2025

Description

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 nondegeneracy 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.

Journal

Bulletin of the London Mathematical Society

Volume

57

Issue

2

First Page

377

Last Page

394

Department

Mathematics

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