{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:LETSJQ4LA2U7N7LGX5IEHZDKZG","short_pith_number":"pith:LETSJQ4L","schema_version":"1.0","canonical_sha256":"592724c38b06a9f6fd66bf5043e46ac9bd65910c1bdbf9924681e0159bc95e72","source":{"kind":"arxiv","id":"1307.1676","version":2},"attestation_state":"computed","paper":{"title":"On the rationality of Poincar\\'e series of Gorenstein algebras via Macaulay's correspondence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Gianfranco Casnati, Joachim Jelisiejew, Roberto Notari","submitted_at":"2013-07-05T17:47:01Z","abstract_excerpt":"Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue field $A/{\\frak M}=k$ of characteristic 0, and let $P_A(z) := \\sum_{p=0}^{\\infty} ({\\mathrm{ Tor}}_p^A(k,k))z^p $ be its Poincar\\'e series. We prove that $P_A(z)$ is rational if either $\\dim_k({{\\frak M}^2/{\\frak M}^3}) \\leq 4 $ and $ \\dim_k(A) \\leq 16,$ or there exist $m\\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\\in [2,c]$ and equal to 1 for $n > c$. The results are obtained thanks to a decomposition of the apolar ideal $\\mathrm {Ann}(F)$ when $F=G+H$ and $G$ and $H$ belong"},"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":"1307.1676","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-07-05T17:47:01Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"284b2fb6dd1619966dc6a36d4e8de6c95d5dbe6b94e0c84a70fd76ef0c52224d","abstract_canon_sha256":"fbf8a8b4aca9a80771918f70b4ff5d94ab352d6b75aed0a95ea15b92aea803a9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:20.828665Z","signature_b64":"zSyZaFMcTiyXX0ifrhCRhONZtjk2Wrpakw2bCEKMTe88DDV94ECxrhH2haR0P27unsdsWHnJZge5BI8Rb7MpAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"592724c38b06a9f6fd66bf5043e46ac9bd65910c1bdbf9924681e0159bc95e72","last_reissued_at":"2026-05-18T03:03:20.828111Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:20.828111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the rationality of Poincar\\'e series of Gorenstein algebras via Macaulay's correspondence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Gianfranco Casnati, Joachim Jelisiejew, Roberto Notari","submitted_at":"2013-07-05T17:47:01Z","abstract_excerpt":"Let $A$ be a local Artinian Gorenstein ring with algebraically closed residue field $A/{\\frak M}=k$ of characteristic 0, and let $P_A(z) := \\sum_{p=0}^{\\infty} ({\\mathrm{ Tor}}_p^A(k,k))z^p $ be its Poincar\\'e series. We prove that $P_A(z)$ is rational if either $\\dim_k({{\\frak M}^2/{\\frak M}^3}) \\leq 4 $ and $ \\dim_k(A) \\leq 16,$ or there exist $m\\leq 4$ and $c$ such that the Hilbert function $H_A(n)$ of $A$ is equal to $ m$ for $n\\in [2,c]$ and equal to 1 for $n > c$. The results are obtained thanks to a decomposition of the apolar ideal $\\mathrm {Ann}(F)$ when $F=G+H$ and $G$ and $H$ belong"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1676","kind":"arxiv","version":2},"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":"1307.1676","created_at":"2026-05-18T03:03:20.828200+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.1676v2","created_at":"2026-05-18T03:03:20.828200+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1676","created_at":"2026-05-18T03:03:20.828200+00:00"},{"alias_kind":"pith_short_12","alias_value":"LETSJQ4LA2U7","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_16","alias_value":"LETSJQ4LA2U7N7LG","created_at":"2026-05-18T12:27:51.066281+00:00"},{"alias_kind":"pith_short_8","alias_value":"LETSJQ4L","created_at":"2026-05-18T12:27:51.066281+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/LETSJQ4LA2U7N7LGX5IEHZDKZG","json":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG.json","graph_json":"https://pith.science/api/pith-number/LETSJQ4LA2U7N7LGX5IEHZDKZG/graph.json","events_json":"https://pith.science/api/pith-number/LETSJQ4LA2U7N7LGX5IEHZDKZG/events.json","paper":"https://pith.science/paper/LETSJQ4L"},"agent_actions":{"view_html":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG","download_json":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG.json","view_paper":"https://pith.science/paper/LETSJQ4L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.1676&json=true","fetch_graph":"https://pith.science/api/pith-number/LETSJQ4LA2U7N7LGX5IEHZDKZG/graph.json","fetch_events":"https://pith.science/api/pith-number/LETSJQ4LA2U7N7LGX5IEHZDKZG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG/action/storage_attestation","attest_author":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG/action/author_attestation","sign_citation":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG/action/citation_signature","submit_replication":"https://pith.science/pith/LETSJQ4LA2U7N7LGX5IEHZDKZG/action/replication_record"}},"created_at":"2026-05-18T03:03:20.828200+00:00","updated_at":"2026-05-18T03:03:20.828200+00:00"}