Cyclic polynomials in Dirichlet-type spaces of the unit bidisk

Abstract For $$\alpha \in \mathbb {R},$$ α ∈ R , we consider the scale of function spaces, namely the Dirichlet-type space $$\mathcal {D}_{\alpha }$$ D α consisting of holomorphic functions on the unit bidisk $$\mathbb {D}^2$$ D 2 , $$f(z,w)=\sum _{k,l=0}^{\infty }a_{kl}z^kw^l$$ f ( z , w ) = ∑ k , l = 0 ∞ a kl z k w l such that $$\begin{aligned} \sum _{k,l=0}^{\infty }(k+l+1)^\alpha |a_{kl}|^2 < \infty . \end{aligned}$$ ∑ k , l = 0 ∞ ( k + l + 1 ) α | a kl | 2 < ∞ . In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial $$2-z_1-z_2$$ 2 - z 1 - z 2 in $$\mathcal {D}_\alpha $$ D α for $$ \frac{3}{2} < \alpha \le 2$$ 3 2 < α ≤ 2 . We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $$\mathcal {D}_\alpha $$