Non-cyclicity and polynomials in Dirichlet-type spaces of the unit ball

TL;DR

This paper characterizes the zero sets of zero-free polynomials in the unit ball of complex n-space and explores conditions for polynomials to be cyclic in Dirichlet-type spaces, proposing a related conjecture.

Contribution

It provides a description of zero sets intersecting the sphere and formulates a new conjecture on polynomial cyclicity in Dirichlet-type spaces.

Findings

Describes the intersection of zero sets with the unit sphere for zero-free polynomials.

Proposes a conjecture on the characterization of cyclic polynomials.

Partially answers the question of non-cyclicity of arbitrary polynomials.

Abstract

We give a description of the intersection of the zero set with the unit sphere of a zero-free polynomial in the unit ball of $\mathbb{C}^n$. This description leads to the formulation of a conjecture regarding the characterization of polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$. Furthermore, we answer partially ascertaining whether an arbitrary polynomial is not cyclic.