Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk
Rajkamal Nailwal, Aljaž Zalar
TL;DR
The paper determines when an irreducible polynomial on the unit bidisk is cyclic in the Dirichlet-type spaces $\mathcal{D}_{\alpha}$. It shows a sharp threshold behavior: for $\alpha\le1$ all zero-free polynomials are cyclic; for $1<\alpha\le2$ cyclicity occurs exactly when the boundary zero set $\mathcal{Z}(p)\cap\mathbb{T}^2$ is empty or finite; and for $\alpha>2$ cyclicity requires that this boundary zero set be empty. The authors develop and combine tools such as slices, diagonal restrictions, and Łojasiewicz-type inequalities, extending prior $\mathfrak{D}_{\alpha}$-space results to the Dirichlet-type setting and addressing the nuanced role of boundary zeros. They also provide a model analysis for the polynomial $2- z_1 - z_2$ and discuss capacity considerations in concluding remarks, highlighting practical criteria for cyclicity in several complex variables.
Abstract
For $α\in \mathbb{R},$ we consider {the scale} of function spaces, namely the Dirichlet-type space ${D}_α$ consisting of holomorphic functions on the unit bidisk $\mathbb{D}^2$, $f(z,w)=\sum_{k,l=0}^{\infty}a_{kl}z^kw^l$ such that $$\sum_{k,l=0}^{\infty}(k+l+1)^α|a_{kl}|^2 < \infty.$$ We present a complete characterization of cyclic polynomials in ${D}_α,$ i.e., given an irreducible polynomial $p,$ the following holds: 1. If $α\leq 1$, then $p$ is cyclic in ${D}_α$. 2. If $1< α\leq 2$, then \( p \) is cyclic in ${D}_α$ if and only if ${Z}(p) \cap \mathbb{T}^2$ is empty or finite. 3. If $α> 2$, then $p$ is cyclic in ${D}_α$ if and only if ${Z}(p) \cap \mathbb{T}^2$ is empty.