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We show that for all $k \\gg 0$ the associated graded ring of the semigroup ring $K[H_{\\mathbf{a}+k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K[H_{\\mathbf{a}+k}]$ itself.\n  As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value dep"},"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":"1308.4644","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-08-21T17:46:44Z","cross_cats_sorted":[],"title_canon_sha256":"f04725d28b3fcb7575cdfdfa929a112e7006194189eea2e87cb4f190cedb4f0a","abstract_canon_sha256":"81afa6ae204db08cd3ca847e565952571ebf728bef84a7e74bc525b10ea02535"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:15.160863Z","signature_b64":"Z6s6WcLboVy+yDKtUmxDvvhoZlPlqk8QSzpD1pJ7RVqICCTbCWqt3RouEa9VqDwubh4ks5Qzyjvb1PUVGiguDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2144ebd5fa57e4e4440f5d4db274979457eb1a741b39a3aa579a1af484dfb8c1","last_reissued_at":"2026-05-18T01:12:15.160519Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:15.160519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the defining equations of the tangent cone of a numerical semigroup ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Dumitru I. Stamate, J\\\"urgen Herzog","submitted_at":"2013-08-21T17:46:44Z","abstract_excerpt":"Let $\\mathbf{a} = a_1 <\\dots < a_r$ be a sequence of positive integers, and let $H_{\\mathbf{a}}$ denote the semigroup generated by $a_1, \\dots, a_r$. For an integer $k\\geq 0$ we denote by $\\mathbf{a}+k$ the shifted sequence $a_1 +k, \\dots, a_r +k$. Fix a field $K$. We show that for all $k \\gg 0$ the associated graded ring of the semigroup ring $K[H_{\\mathbf{a}+k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K[H_{\\mathbf{a}+k}]$ itself.\n  As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value dep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4644","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":"1308.4644","created_at":"2026-05-18T01:12:15.160579+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.4644v3","created_at":"2026-05-18T01:12:15.160579+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.4644","created_at":"2026-05-18T01:12:15.160579+00:00"},{"alias_kind":"pith_short_12","alias_value":"EFCOXVP2K7SO","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_16","alias_value":"EFCOXVP2K7SOIRAP","created_at":"2026-05-18T12:27:43.054852+00:00"},{"alias_kind":"pith_short_8","alias_value":"EFCOXVP2","created_at":"2026-05-18T12:27:43.054852+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/EFCOXVP2K7SOIRAPLVG3E5EXSR","json":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR.json","graph_json":"https://pith.science/api/pith-number/EFCOXVP2K7SOIRAPLVG3E5EXSR/graph.json","events_json":"https://pith.science/api/pith-number/EFCOXVP2K7SOIRAPLVG3E5EXSR/events.json","paper":"https://pith.science/paper/EFCOXVP2"},"agent_actions":{"view_html":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR","download_json":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR.json","view_paper":"https://pith.science/paper/EFCOXVP2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.4644&json=true","fetch_graph":"https://pith.science/api/pith-number/EFCOXVP2K7SOIRAPLVG3E5EXSR/graph.json","fetch_events":"https://pith.science/api/pith-number/EFCOXVP2K7SOIRAPLVG3E5EXSR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR/action/storage_attestation","attest_author":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR/action/author_attestation","sign_citation":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR/action/citation_signature","submit_replication":"https://pith.science/pith/EFCOXVP2K7SOIRAPLVG3E5EXSR/action/replication_record"}},"created_at":"2026-05-18T01:12:15.160579+00:00","updated_at":"2026-05-18T01:12:15.160579+00:00"}