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Cyclic polynomials in anisotropic Dirichlet~spaces

Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions $f(z_1,z_2):=\sum_{k,l\geq 0}a_{kl}z_1^kz_2^l$ such that $\sum_{k,l\geq 0}(k+1)^{α_1} (l+1)^{α_2}|a_{kl}|^2 <\infty.$ Here the parameters $α_1,α_2$ 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 $α_1+α_2\leq 1$, then $p$ is cyclic; if $α_1+α_2>1$ and $\min\{α_1,α_2\}\leq 1$, then $p$ is cyclic if and only if it has finitely many zeros in the two-torus $\mathbb T^2$; if $\min\{α_1,α_2\}>1$, then $p$ is cyclic if and only if it has no zeros in $\mathbb T^2$.