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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1512.04871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2015-12-15T17:29:45Z","cross_cats_sorted":[],"title_canon_sha256":"9e618f54e333d7e34157217eeb6d52acfd70754070afdbc19ad3ea6af4740e52","abstract_canon_sha256":"145ba0fe7941bda0e7ab503327025ccb7c16222c86e9bc7b0dfea01ea12a9639"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:15.211272Z","signature_b64":"lSW12rW0WAt6bO9daWBXh0D+NjMkHi4YUPi6J1suYXbhTG4pZL/zFN4jThDRjmacp5KxSt9ywg/yi/nu1W3dDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc653ae09645a6f2eb7ee657015029b0e8a90f4dbe5784bf5e92e92eb46d5d41","last_reissued_at":"2026-05-18T01:24:15.210643Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:15.210643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1512.04871","created_at":"2026-05-18T01:24:15.210709+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.04871v1","created_at":"2026-05-18T01:24:15.210709+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.04871","created_at":"2026-05-18T01:24:15.210709+00:00"},{"alias_kind":"pith_short_12","alias_value":"7RSTVYEWIWTP","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7RSTVYEWIWTPF236","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7RSTVYEW","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD","json":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD.json","graph_json":"https://pith.science/api/pith-number/7RSTVYEWIWTPF2364ZLQCUBJWD/graph.json","events_json":"https://pith.science/api/pith-number/7RSTVYEWIWTPF2364ZLQCUBJWD/events.json","paper":"https://pith.science/paper/7RSTVYEW"},"agent_actions":{"view_html":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD","download_json":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD.json","view_paper":"https://pith.science/paper/7RSTVYEW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.04871&json=true","fetch_graph":"https://pith.science/api/pith-number/7RSTVYEWIWTPF2364ZLQCUBJWD/graph.json","fetch_events":"https://pith.science/api/pith-number/7RSTVYEWIWTPF2364ZLQCUBJWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD/action/storage_attestation","attest_author":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD/action/author_attestation","sign_citation":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD/action/citation_signature","submit_replication":"https://pith.science/pith/7RSTVYEWIWTPF2364ZLQCUBJWD/action/replication_record"}},"created_at":"2026-05-18T01:24:15.210709+00:00","updated_at":"2026-05-18T01:24:15.210709+00:00"}