{"paper":{"title":"Reduced Gr\\\"obner Bases and Macaulay-Buchberger Basis Theorem over Noetherian Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"cs.SC","authors_text":"Ambedkar Dukkipati, Maria Francis","submitted_at":"2013-04-25T12:14:49Z","abstract_excerpt":"In this paper, we extend the characterization of $\\mathbb{Z}[x]/\\ < f \\ >$, where $f \\in \\mathbb{Z}[x]$ to be a free $\\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows us to extend the Gr\\\"obner basis method of computing a $\\Bbbk$-vector space basis of residue class polynomial rings over a field $\\Bbbk$ (Macaulay-Buchberger Basis Theorem) to rings, i.e. $A[x_1,\\ldots,x_n]/\\mathfrak{a}$, where $\\mathfrak{a} \\subseteq A[x_1,\\ldots,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.6889","kind":"arxiv","version":5},"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"}