{"paper":{"title":"Standard Bases in mixed Power Series and Polynomial Rings over Rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Oliver Wienand, Thomas Markwig, Yue Ren","submitted_at":"2015-09-24T20:25:42Z","abstract_excerpt":"In this paper we study standard bases for submodules of a mixed power series and polynomial ring $R[[t_1,\\ldots,t_m]][x_1,\\ldots,x_n]^s$ respectively of their localization with respect to a $t$-local monomial ordering for a certain class of noetherian rings $R$. The main steps are to prove the existence of a division with remainder generalizing and combining the division theorems of Grauert--Hironaka and Mora and to generalize the Buchberger criterion. Everything else then translates naturally. Setting either $m=0$ or $n=0$ we get standard bases for polynomial rings respectively for power seri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07528","kind":"arxiv","version":2},"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"}