{"paper":{"title":"Neighbors, Generic Sets and Scarf-Buchberger Hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"James J. Madden, Trevor McGuire","submitted_at":"2015-11-25T21:03:40Z","abstract_excerpt":"The present paper is motivated by the need to generalize the construction of the Scarf complex in order to give combinatorial resolutions of a much broader class of modules than just the monomial ideals. For any subset $A\\subseteq \\mathbb{R}^n$, let $\\mathfrak{N}(A)$ denote the collection of all subsets $B\\subseteq A$ such that there is no $a\\in A$ that is strictly less than the supremum of $B$ in all coordinates. We show that if $A\\subseteq \\mathbb{Z}^n$ is generic (in a sense appropriate for this context), then $\\mathfrak{N}(A)$ is a locally finite simplicial complex. Moreover, if $A$ is gen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08224","kind":"arxiv","version":1},"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"}