2015/09/29 by Thomas Markwig, Yue Ren, Oliver Wienand
Computer Science · Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Arithmetic #Class (philosophy) #Commutative Algebra and Its Applications #Computer science #Discrete mathematics #Division (mathematics) #Finitely-generated abelian group #Mathematical analysis #Mathematics #Monomial #Noetherian #Noetherian ring #Polynomial #Polynomial and algebraic computation #Polynomial ring #Power series #Pure mathematics #Ring (chemistry) #Series (stratigraphy) #math.AG #msc:12J25 #msc:13F25 #msc:13P10 #msc:16W60
paper · pdf · doi:10.1016/j.jsc.2016.08.009
published as Journal of Symbolic Computation, 79 (2017), pp. 119-139 · 44 pages
arxiv created 2015/09/29 · openalex publication_date 2016/08/31 · arxiv updated 2016/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper we study standard bases for submodules of a mixed power series and polynomial ring R[[t1,…,tm]][x1,…,xn]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 series rings over R as a special case.