{"paper":{"title":"On Gr\\\"obner Bases and Krull Dimension of Residue Class Rings of Polynomial Rings over Integral Domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"Ambedkar Dukkipati, Maria Francis","submitted_at":"2016-02-13T08:11:02Z","abstract_excerpt":"Given an ideal $\\mathfrak{a}$ in $A[x_1, \\ldots, x_n]$, where $A$ is a Noetherian integral domain, we propose an approach to compute the Krull dimension of $A[x_1,\\ldots,x_n]/\\mathfrak{a}$, when the residue class polynomial ring is a free $A$-module. When $A$ is a field, the Krull dimension of $A[x_1,\\ldots,x_n]/\\mathfrak{a}$ has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. For a Noetherian integral domain, $A$ we introduce the notion of combinatorial dimension of $A[x_1, \\ldots,x_n]/\\mathfrak{a}$ and gi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04300","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"}