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In this paper we study the length of generalized fractions $M (1/(x_1, ..., x_d, 1))$ which was introduced by Sharp and Hamieh in \\cite{ShH85}. First, we study the growth of the function $J_{\\underline{x}, M}(\\underline{n}) = \\ell(M (1/(x_1^{n_1}, ..., x_d^{n_d}, 1))) - n_1...n_d e(\\underline{x};M)$. Then we give an explicit calculation for the function"},"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":"1405.7240","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-05-28T13:54:48Z","cross_cats_sorted":[],"title_canon_sha256":"37b64e879d65f4105d18ed45ad151380106df9c2a0b79a2a72c126624f2bca69","abstract_canon_sha256":"46985c81aa63075d6da8d7c77f1ddc87cf5ccd3f8d3a5c8fb88d4ce593e0d7bb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:58.292662Z","signature_b64":"2kBaz57v84htu3CvDizAKAIPJL0lkKvV5pZhVGMRJHN/NrWg8XkK4aVfuB3PsR2yhLnGHplKMfOOdufqahE0AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2bac03e6ee235a7b89f315db617145dc252cce53193728d42f6d6927ccf19f78","last_reissued_at":"2026-05-18T02:50:58.291971Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:58.291971Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A study of the length function of generalized fractions of modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Marcel Morales, Pham Hung Quy","submitted_at":"2014-05-28T13:54:48Z","abstract_excerpt":"Let $(R, \\frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. Let $\\underline{x} = x_1, ..., x_d$ be a system of parameters of $M$ and $\\underline{n} = (n_1, ..., n_d)$ a $d$-tuple of positive integers. In this paper we study the length of generalized fractions $M (1/(x_1, ..., x_d, 1))$ which was introduced by Sharp and Hamieh in \\cite{ShH85}. First, we study the growth of the function $J_{\\underline{x}, M}(\\underline{n}) = \\ell(M (1/(x_1^{n_1}, ..., x_d^{n_d}, 1))) - n_1...n_d e(\\underline{x};M)$. 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