Cyclic polynomials in two variables (Q2822718)

Let \(\alpha\in \mathbb{R}\). Then the Dirichlet-type space, \(\mathcal{D}_{\alpha}\), is the space of holomorphic functions \(f\) on the bidisk \(\mathbb{D}^{2}\) of \(\mathbb{C}^{2}\) with Taylor coefficients \(\{a_{k,l}\}_{k,l\in \mathbb{N}^{2}}\) in the Taylor expansion NEWLINE\[NEWLINEf(z,w)=\sum_{k=0}\sum_{l=0}^{\infty}a_{k,l}z^{k}w^{l}NEWLINE\]NEWLINE such that NEWLINE\[NEWLINE\sum_{k=0}\sum_{l=0}^{\infty}(k+1)^{\alpha}(l+1)^{\alpha}|a_{k,l}|^{2}<\infty.NEWLINE\]NEWLINE The authors give a complete characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shift acting on \(\mathcal{D}_{\alpha}\), which includes the Hardy space and the Dirichlet space of the bidisk. For this define \(\mathbb{T}^{2}:=\{(z,w)\in \mathbb{C}^{2} : |z|=|w|=1\}\). The main result in this paper is the following:NEWLINENEWLINETheorem. Let \(f(z,w)\) be an irreducible polynomial with no zero in the bidisk. DenoteNEWLINENEWLINE \(\mathcal{Z}(f):=\{(z,w)\in \mathbb{C}^{2} : f(z,w)=0\}\). {\parindent=0.6cm\begin{itemize}\item[(1)] If \(\alpha\leq 1/2\), then \(f\) is cyclic in \(\mathcal{D}_{\alpha}\). \item[(2)] If \(\alpha\in (1/2, 1]\), then \(f\) is cyclic in \(\mathcal{D}_{\alpha}\) if and only if one of the following holds: {\parindent=0.8cm\begin{itemize}\item[(a)] \(\mathcal{Z}(f) \cap \mathbb{T}^{2}\) is empty or a finite set. \item[(b)] \(f\) is a constant multiple of \(\xi-z\) or \(\xi-w\) for some \(\xi\in \mathbb{T}\). NEWLINENEWLINE\end{itemize}} \item[(3)] If \(\alpha>1\), then \(f\) is cyclic in \(\mathcal{D}_{\alpha}\) if and only if \(\mathcal{Z}(f)\cap \mathbb{T}^{2}\) is empty. NEWLINENEWLINE\end{itemize}}