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Next, we prove a simple numerical criterion for $R$ to be Cohen-Macaulay in the case when $t=2$. We also provide a simple algorithm which identifies the monomial $k$-basis of $R/(x^a,y^b)$. Finally, these simple results are specialized to the case of projective monomial curves in $\\mathbb{P}^3$."},"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":"1602.05843","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2016-02-18T15:45:04Z","cross_cats_sorted":[],"title_canon_sha256":"bb1acaf38c30f659579868f3f7ecd413a0eaec3096356de117eae0f3699741bd","abstract_canon_sha256":"37adf6cdd1be3239f211ed3c2bff10526c84ff53f104ca7da1b42b87566f5798"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:23.080735Z","signature_b64":"+VIhn5wRmOA1UWNO95KxtGrm6KmHR99781DzW3ilr5kD1xWXFvlis5py8RjRjM9te//Z2cdcFOppFChxHZRiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f21634ff16728cc85243b3c6fdefc066c3a4cac6e437d330a986aa1c20e148e","last_reissued_at":"2026-05-18T01:20:23.080041Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:23.080041Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Cohen-Macaulay Property of Affine Semigroup Rings in Dimension 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Grant Serio, Tony Se","submitted_at":"2016-02-18T15:45:04Z","abstract_excerpt":"Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\\ldots,x^{p_t}y^{s_t},y^b] \\subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion for the ring to be Cohen-Macaulay. Next, we prove a simple numerical criterion for $R$ to be Cohen-Macaulay in the case when $t=2$. We also provide a simple algorithm which identifies the monomial $k$-basis of $R/(x^a,y^b)$. 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