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We study the singularities of C by studying a Hilbert-Burch matrix phi for the row vector [g_1,...,g_n]. In the \"General Lemma\" we use the generalized row ideals of phi to identify the singular points on C, their multiplicities, the number of branches at each singular point, and the multiplicity of each branch.\n  Let p be a singular point on the parameterized planar curve C which"},"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":"1102.5072","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-02-24T19:43:36Z","cross_cats_sorted":[],"title_canon_sha256":"4f5b45cb29bd7afabe017a5929ebe55c1a5e2355928a11fdf2f087fa553027ca","abstract_canon_sha256":"770bc3ff1f0192de407b3fdb1e1c915daa0e4a48488a5276bdb45aabd9e0eeec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:52.849584Z","signature_b64":"qdf7Kbh/jnD8XTVM+EjCoCKNWFJcbJQIjqOd0ZSPpjkp92q/guPRfzLDfCOMYlZlbaVlKzSoLKrZdSx1VnmTDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63c52136639460eafa79503eeedb4ae45fd850f7467e879919e63f7820fec246","last_reissued_at":"2026-05-18T04:02:52.848682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:52.848682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A study of singularities on rational curves via syzygies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Andrew R. 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