Reduction of everywhere convergent power series with respect to Gröbner bases
type: Item
instance of: scholarly article
title: Reduction of everywhere convergent power series with respect to Gröbner bases
zbMATH Open document ID: 0859.13015
DOI: 10.1016/0022-4049(95)00102-6
author: Joachim Apel
author: J. Stückrad
author: T. Winiarski
author: Piotr Tworzewski
published in: Journal of Pure and Applied Algebra
publication date: 1997-04-10T00:00:00Z
full work available at URL: qucosa:34418
review text: The formula for the division by a finite set \(F\) of polynomials of several variables \[ g=\sum_{f\in F}h_f\cdot f+g_{\text{red}} \] is generalized to the case of the ring \(E\) of everywhere convergent power series (\(g\), \(h_f\), \(g_{\text{red}}\) are power series then and, for a suitable norm \(|\cdot |_r\), \(|g_{\text{red}} |_r\leq|g|_r\)). The authors point out that the similar topic was investigated earlier by \textit{P. D. Djakov} and \textit{B. S. Mitiagin} [Stud. Math. 68, 85-104 (1980; Zbl 0434.46034)]. In the present article, however, Gröbner bases are used and, in the case, where \(F\) is the Gröbner basis of a polynomial ideal, uniqueness of \(g_{\text{red}}\), in a certain sense, is established. The used technique is applied to the proof of closedness of ideals generated by polynomials in \(E\) and to a simple proof for the affine version of \textit{J. P. Serre}'s graph theorem [GAGA, Ann. Inst. Fourier 6(1955/1956), 1-42 (1956; Zbl 0075.30401)].
Mathematics Subject Classification ID: 13J05
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 896569
zbMATH Keywords: division of convergent power series
zbMATH Keywords: Gröbner bases
zbMATH Keywords: polynomial ideal
MaRDI profile type: Publication
OpenAlex ID: W1966338446
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description: scientific article; zbMATH DE number 896569