Cyclicity in Poletsky-Stessin Weighted Bergman Spaces

TL;DR

This paper investigates the cyclicity of polynomials in Poletsky-Stessin weighted Bergman spaces on various complex domains, extending parameter ranges and comparing behaviors across different spaces, with implications for understanding their structure.

Contribution

It introduces an extended parameter range for Poletsky-Stessin weighted Bergman spaces on complete Reinhardt domains and compares cyclicity behavior across different complex domains.

Findings

Differences in polynomial cyclicity between bidisk and other domains.

Extension of parameter ranges for Poletsky-Stessin spaces.

Open problems on cyclic polynomial structures.

Abstract

We study the cyclicity of polynomials in Poletsky-Stessin weighted Bergman spaces on various domains in $\mathbb{C}^2$, including the unit ball, the bidisk, and the complex ellipsoid. To this end, we introduce a natural extension of the parameter range for Poletsky-Stessin weighted Bergman spaces on complete Reinhardt domains, yielding a family of spaces that resemble Dirichlet-type spaces on the unit ball. We highlight the differences in the cyclicity behavior of polynomials in these spaces on the bidisk compared to those studied by B\'en\'eteau et al. Finally, we propose several open problems concerning the structure of cyclic polynomials in these spaces.