Cyclic polynomials in two variables

TL;DR

This paper characterizes cyclic polynomials in two complex variables within Dirichlet-type spaces on the bidisk, linking cyclicity to the polynomial's zero set and employing advanced techniques from analysis and harmonic analysis.

Contribution

It provides a complete characterization of cyclic polynomials in two variables for Dirichlet-type spaces, connecting cyclicity to boundary zero set properties.

Findings

Cyclicity depends on the size and nature of the zero set on the boundary.

The characterization applies to Hardy and Dirichlet spaces of the bidisk.

Techniques include real analytic functions, determinantal representations, and harmonic analysis.

Abstract

We give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for stable polynomials, and harmonic analysis on curves