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Suppose that $L$ is the lattice of integers points of $(a_1,\\dots,a_n)^{\\perp}$. Taking cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules $M_L^{(k)}$ whose Castelnuovo-Mumford regularity captures the $k$-th Frobenius number of $(a_1,\\dots,a_n)$. We study th"},"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":"1703.10884","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-03-31T13:01:22Z","cross_cats_sorted":[],"title_canon_sha256":"757aa79f670b3d0558a3e70a97ee4568616fb61ad92b7a9de2e1c350c2a6f5bf","abstract_canon_sha256":"f85404e262c4076b07dbd10b7db2b35233a662d1914101845349e99d378fabd1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:47.830048Z","signature_b64":"AS+z2L9uv6fcTvkDJ2XESV7pQWMdYYvq6jtDC/fNB8plEiK6Gz/32qYI1NcFGYkeBnQh3EwDtkF6el8ATjJtCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"465b3312f12062e573096598d028920b21d763783006d2db25cf368e444629df","last_reissued_at":"2026-05-18T00:10:47.829377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:47.829377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutative Algebra of Generalised Frobenius Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Ben Smith, Madhusudan Manjunath","submitted_at":"2017-03-31T13:01:22Z","abstract_excerpt":"We study commutative algebra arising from generalised Frobenius numbers. The $k$-th (generalised) Frobenius number of natural numbers $(a_1,\\dots,a_n)$ is the largest natural number that cannot be written as a non-negative integral combination of $(a_1,\\dots,a_n)$ in $k$ distinct ways. Suppose that $L$ is the lattice of integers points of $(a_1,\\dots,a_n)^{\\perp}$. Taking cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules $M_L^{(k)}$ whose Castelnuovo-Mumford regularity captures the $k$-th Frobenius number of $(a_1,\\dots,a_n)$. 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