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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300888&_mathId=si1.gif&_user=111111111&_pii=S0747717116300888&_rdoc=1&_issn=07477171&md5=cee40627550e37aea8491600afdc46a1" title="Click to view the MathML source">R〚t₁,…,t_m〛[x₁,…,x_n]^sclass="mathContainer hidden">class="mathCode">$R〚t_1,\dots ,t_m〛{[x_1,\dots ,x_n]}^s$ respectively of their localisation with respect to a class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300888&_mathId=si2.gif&_user=111111111&_pii=S0747717116300888&_rdoc=1&_issn=07477171&md5=87c9ab6319975b5697d59849513e5439">class="imgLazyJSB inlineImage" height="11" width="8" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0747717116300888-si2.gif">class="mathContainer hidden">class="mathCode">$\underset{\_}{t}$-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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300888&_mathId=si3.gif&_user=111111111&_pii=S0747717116300888&_rdoc=1&_issn=07477171&md5=06916aae867e73c734cee90592cb543b" title="Click to view the MathML source">m=0class="mathContainer hidden">class="mathCode">$m=0$ or class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300888&_mathId=si4.gif&_user=111111111&_pii=S0747717116300888&_rdoc=1&_issn=07477171&md5=1dd7d530e242c36d0d77082f13993b41" title="Click to view the MathML source">n=0class="mathContainer hidden">class="mathCode">$n=0$ we get standard bases for polynomial rings respectively for power series rings over R as a special case.