Cyclic polynomials in Dirichlet-type spaces of the unit bidisk

For \(\alpha \in \mathbb {R},\) we consider the scale of function spaces, namely the Dirichlet-type space \(\mathcal {D}_{\alpha }\) 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

$$\begin{aligned} \sum _{k,l=0}^{\infty }(k+l+1)^\alpha |a_{kl}|^2 < \infty . \end{aligned}$$

In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial \(2-z_1-z_2\) in \(\mathcal {D}_\alpha \) for \( \frac{3}{2} < \alpha \le 2\). We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in \(\mathcal {D}_\alpha \).