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Stable polynomials and bounded rational functions in the unit ball

Greg Knese, James Eldred Pascoe, Alan Sola

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.

Stable polynomials and bounded rational functions in the unit ball

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.
Paper Structure (15 sections, 18 theorems, 208 equations, 1 figure)

This paper contains 15 sections, 18 theorems, 208 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Simple zero on the boundary
  3. Two variables
  4. Basic cases
  5. Admissible numerators in basic case
  6. The non-basic cases: isolated and curve type
  7. Admissible numerators for isolated type
  8. Global examples
  9. Elementary examples
  10. Stable polynomials via one-variable polynomials
  11. Examples from quadratic forms
  12. Rudin's construction
  13. Stable polynomials via row contractions
  14. Examples beyond the ball setting
  15. Constructing polynomials with local behavior

Key Result

Theorem 1.2

Let $p \in \mathbb{C}[z_1,\dots, z_{d-1},w]$ be non-vanishing in $U_d$ near $0$, $p(0) = 0$, and $\nabla p(0) \ne 0$. Let $Hp(0) = \left(\frac{\partial^2 p}{\partial z_j z_k}(0)\right)_{j,k=1,\dots,d-1}$ be the Hessian matrix of $p$ with respect to $z$ at $0$. Then, $\nabla p(0)$ is a multiple of $\ is contractive. For a partial converse, if $p$ is a polynomial such that $p(0)=0$, $\nabla p(0)$ is

Figures (1)

  • Figure 1: Bernoulli Lemniscate

Theorems & Definitions (47)

  • Example 1.1
  • Theorem 1.2
  • Proposition 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 37 more