{"paper":{"title":"On the defining equations of the tangent cone of a numerical semigroup ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Dumitru I. Stamate, J\\\"urgen Herzog","submitted_at":"2013-08-21T17:46:44Z","abstract_excerpt":"Let $\\mathbf{a} = a_1 <\\dots < a_r$ be a sequence of positive integers, and let $H_{\\mathbf{a}}$ denote the semigroup generated by $a_1, \\dots, a_r$. For an integer $k\\geq 0$ we denote by $\\mathbf{a}+k$ the shifted sequence $a_1 +k, \\dots, a_r +k$. Fix a field $K$. We show that for all $k \\gg 0$ the associated graded ring of the semigroup ring $K[H_{\\mathbf{a}+k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K[H_{\\mathbf{a}+k}]$ itself.\n  As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value dep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4644","kind":"arxiv","version":3},"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"}