{"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 $(a_1,\\dots,a_n)$. We study th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10884","kind":"arxiv","version":2},"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"}