Standard bases in mixed power series and polynomial rings over rings

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• 作者：Thomas Markwig^a ; ^{keilen@math.uni-tuebingen.de" class="auth_mail" title="E-mail the corresponding author} ; Yue Ren^b ; ^{ren@mathematik.uni-kl.de" class="auth_mail" title="E-mail the corresponding author} ; Oliver Wienand^b ; ^{wienand@mathematik.uni-kl.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13P10 ; 13F25 ; 16W60 ; secondary ; 12J25 ; 16W60
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P1
• 页码：119-139
• 全文大小：468 K

文摘

In this paper we study standard bases for submodules of a mixed power series and polynomial ring 627550e37aea8491600afdc46a1" title="Click to view the MathML source">R〚t₁,…,t_m〛[x₁,…,x_n]^s$R〚t_1,\dots ,t_m〛{[x_1,\dots ,x_n]}^s$ respectively of their localisation with respect to a $\underset{\_}{t}$-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$m=0$ or 2f13993b41" title="Click to view the MathML source">n=0$n=0$ we get standard bases for polynomial rings respectively for power series rings over R as a special case.