{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:S5Y67MVCILCUER5EDL7O632G4S","short_pith_number":"pith:S5Y67MVC","schema_version":"1.0","canonical_sha256":"9771efb2a242c54247a41afeef6f46e4bccfec5fc5a670fe36ce7a2c3432460f","source":{"kind":"arxiv","id":"1703.09427","version":3},"attestation_state":"computed","paper":{"title":"The multiplicity and the number of generators of an integrally closed ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Hailong Dao, Ilya Smirnov","submitted_at":"2017-03-28T07:28:19Z","abstract_excerpt":"Let $(R, \\mathfrak m)$ be a Noetherian local ring and $I$ a $\\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay"},"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":"1703.09427","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-28T07:28:19Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"d21e844973bebc164351154b40052d1529c7f747024061b2fb98637a4e00af3e","abstract_canon_sha256":"3436d995c9356b9e26e0df668715477cc4ab4e79b64d1dc80d10b4c8f2dc608a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:56.679064Z","signature_b64":"tq0vf+n0qVui/aR4TJqgUUBh4pI00UKeoEyWh0bnKyzA+Ay4WwPSWSg85MQ0nsb/qKsFoiIMbmmILd1kStV+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9771efb2a242c54247a41afeef6f46e4bccfec5fc5a670fe36ce7a2c3432460f","last_reissued_at":"2026-05-18T00:23:56.678465Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:56.678465Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The multiplicity and the number of generators of an integrally closed ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Hailong Dao, Ilya Smirnov","submitted_at":"2017-03-28T07:28:19Z","abstract_excerpt":"Let $(R, \\mathfrak m)$ be a Noetherian local ring and $I$ a $\\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09427","kind":"arxiv","version":3},"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":"1703.09427","created_at":"2026-05-18T00:23:56.678559+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.09427v3","created_at":"2026-05-18T00:23:56.678559+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.09427","created_at":"2026-05-18T00:23:56.678559+00:00"},{"alias_kind":"pith_short_12","alias_value":"S5Y67MVCILCU","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_16","alias_value":"S5Y67MVCILCUER5E","created_at":"2026-05-18T12:31:43.269735+00:00"},{"alias_kind":"pith_short_8","alias_value":"S5Y67MVC","created_at":"2026-05-18T12:31:43.269735+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/S5Y67MVCILCUER5EDL7O632G4S","json":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S.json","graph_json":"https://pith.science/api/pith-number/S5Y67MVCILCUER5EDL7O632G4S/graph.json","events_json":"https://pith.science/api/pith-number/S5Y67MVCILCUER5EDL7O632G4S/events.json","paper":"https://pith.science/paper/S5Y67MVC"},"agent_actions":{"view_html":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S","download_json":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S.json","view_paper":"https://pith.science/paper/S5Y67MVC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.09427&json=true","fetch_graph":"https://pith.science/api/pith-number/S5Y67MVCILCUER5EDL7O632G4S/graph.json","fetch_events":"https://pith.science/api/pith-number/S5Y67MVCILCUER5EDL7O632G4S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S/action/storage_attestation","attest_author":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S/action/author_attestation","sign_citation":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S/action/citation_signature","submit_replication":"https://pith.science/pith/S5Y67MVCILCUER5EDL7O632G4S/action/replication_record"}},"created_at":"2026-05-18T00:23:56.678559+00:00","updated_at":"2026-05-18T00:23:56.678559+00:00"}