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 ${\mathbb{D}}^2$, $f(z,w)={\sum }_{k,l=0}^{\mathrm{\infty }}{a}_{kl}{z}^k{w}^l$ such that $$\sum _{k,l=0}^{\mathrm{\infty }}(k+l+1{)}^{\alpha }\mathrm{\mid }{a}_{kl}{\mathrm{\mid }}^2<\mathrm{\infty }\mathrm{.}$$ 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.