Cyclic polynomials in anisotropic Dirichlet~spaces

Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions $f(z_1,z_2):={\sum }_{k,l\ge 0}{a}_{kl}{z}_1^k{z}_2^l$ such that ${\sum }_{k,l\ge 0}(k+1{)}^{{\alpha }_1}(l+1{)}^{{\alpha }_2}\mathrm{\mid }{a}_{kl}{\mathrm{\mid }}^2<\mathrm{\infty }\mathrm{.}$ Here the parameters ${\alpha }_1,{\alpha }_2$ are arbitrary real numbers. We characterize the polynomials that are cyclic for the shift operators on this space. More precisely, we show that, given an irreducible polynomial $p(z_1,z_2)$ depending on both $z_1$ and $z_2$ and having no zeros in the bidisk: if ${\alpha }_1+{\alpha }_2\le 1$, then $p$ is cyclic; if ${\alpha }_1+{\alpha }_2>1$ and $\min \{{\alpha }_1,{\alpha }_2\}\le 1$, then $p$ is cyclic if and only if it has finitely many zeros in the two-torus ${\mathbb{T}}^2$; if $\min \{{\alpha }_1,{\alpha }_2\}>1$, then $p$ is cyclic if and only if it has no zeros in ${\mathbb{T}}^2$.