{"paper":{"title":"Efficient Gr\\\"obner Bases Computation over Principal Ideal Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Christian Eder, Tommy Hofmann","submitted_at":"2019-06-20T10:36:54Z","abstract_excerpt":"In this paper we present a new efficient variant to compute strong Gr\\\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$. Possibly using available factorization algorithms we may thus recursively reduce some strong Gr\\\"obner basis computations to Gr\\\"obner basis computations over fields for prime factors of $n$, at least for squarefree $n$. Considering now a computation over $R/nR$ we can run a standard Gr\\\"obner b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08543","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"}