Standard bases in mixed power series and polynomial rings over rings. (March 2017)

• Record Type: Journal Article
• Authors: Markwig, Thomas; Ren, Yue; Wienand, Oliver
• Is Part Of: Journal of symbolic computation. Volume 79(2017)Part 1
• Page Start: 119
• Page End: 139
• Publication Date: 2017-03
• Subjects: primary 13P10 13F25 16W60 -- secondary 12J25 16W60; Standard basis -- Monomial ordering -- Division with remainder; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.08.009
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper we study standard bases for submodules of a mixed power series and polynomial ring R 〚 t 1, …, t m 〛 [ x 1, …, x n ] s respectively of their localisation with respect to a t _ -local monomial ordering for a certain class of noetherian rings R, also called Zacharias rings. The main steps are to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora and to generalise 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.