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 $$\sum_{k,l=0}^{\infty}(k+l+1)^\alpha|a_{kl}|^2 < \infty.$$ In this paper, we solve an open problem posed in \cite[Open problem~1]{Z25}, which asks whether the polynomial $2 - z_1 - z_2$ is cyclic 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 $\mathcal{D}_\alpha$. In addition, we establish several properties of cyclic functions and present alternative proofs of some cases previously obtained by P. T. Ziarati.