{"paper":{"title":"Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.ST","stat.TH"],"primary_cat":"math.AC","authors_text":"Erik Stokes, Sonja Petrovi\\'c","submitted_at":"2009-10-08T20:46:09Z","abstract_excerpt":"There are two seemingly unrelated ideals associated with a simplicial complex \\Delta. One is the Stanley-Reisner ideal I_\\Delta, the monomial ideal generated by minimal non-faces of \\Delta, well-known in combinatorial commutative algebra. The other is the toric ideal I_{M(\\Delta)} of the facet subring of \\Delta, whose generators give a Markov basis for the hierarchical model defined by \\Delta, playing a prominent role in algebraic statistics.\n  In this note we show that the complexity of the generators of I_{M(\\Delta)} is determined by the Betti numbers of I_\\Delta. The unexpected connection b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.1610","kind":"arxiv","version":4},"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"}