# Standard Bases in mixed Power Series and Polynomial Rings over Rings

Abstract : 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 series rings over $R$ as a special case.
Document type :
Conference papers

https://hal.inria.fr/hal-01350996
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Submitted on : Tuesday, August 2, 2016 - 2:16:02 PM
Last modification on : Thursday, October 28, 2021 - 3:42:03 PM

### Identifiers

• HAL Id : hal-01350996, version 1
• ARXIV : 1509.07528

### Citation

Thomas Markwig, yue Ren, Oliver Wienand. Standard Bases in mixed Power Series and Polynomial Rings over Rings. MEGA'2015 (Special Issue), Jun 2015, Trento, Italy. ⟨hal-01350996⟩

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