{"paper":{"title":"Cyclic polynomials in anisotropic Dirichlet~spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Alan Sola, Greg Knese, Lukasz Kosinski, Thomas J. Ransford","submitted_at":"2015-12-15T17:29:45Z","abstract_excerpt":"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 $\\sum_{k,l\\geq 0}(k+1)^{\\alpha_1} (l+1)^{\\alpha_2}|a_{kl}|^2 <\\infty.$ 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 $\\alpha_1+\\alpha_2\\leq 1$, then $p$ is cyclic; if $\\alpha_1+\\alpha_2>1$ and $\\min\\{\\al"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04871","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}