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 ${D}_{\alpha}$ 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 $$\sum_{k,l=0}^{\infty}(k+l+1)^\alpha|a_{kl}|^2 < \infty.$$ In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial $2-z_1-z_2$ in $ D_\alpha$ for $ \frac32 < \alpha \leq 2$.

We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $ D_\alpha$.