Adapting coordinates
type: Item
instance of: scholarly article
title: Adapting coordinates
zbMATH Open document ID: 1028.13005
DOI: 10.1016/S0022-4049(03)00070-7
author: Joost Berson
author: Jan Willem Bikker
author: Arno van den Essen
published in: Journal of Pure and Applied Algebra
publication date: 2003-10-15T00:00:00Z
review text: Let \(A\) be a commutative ring. A polynomial \(f\in A[x_1,\dots,x_n]\) is called a coordinate iff there exist \(f_2,\dots,f_n \in A[x_1,\dots,x_n]\) such that \(A[f,f_2,\dots,f_n]=A[x_1,\dots,x_n]\). The problem considered in the paper is to find criteria for \(f\) to be a coordinate (in the particular case when \(A\) is a field, e.g. \(A=\mathbb C\), this problem is connected to the jacobian conjecture and has been investigated by many authors). The results of the paper extend and generalize work by \textit{E. Edo} and \textit{S. Vénéreau} [Ann. Pol. Math. 76, 67-76 (2001; Zbl 0989.13013)]. The main results are some sufficient conditions for \(f\) to be a coordinate in terms of localizations and quotients of \(A\). We quote one of them in the two-dimensional case (\(n=2\)). Let \(A\) be a \(\mathbb Q\)-algebra, \(a\in A\) and \(f\in A[y,z]\). If \(f\) is a coordinate in the rings \(A_a[y,z]\) and \((A/Aa)[y,z]\), then \(f\) is a coordinate in \(A[y,z]\).
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14R15
zbMATH DE Number: 1993281
zbMATH Keywords: jacobian conjecture
zbMATH Keywords: polynomial automorphism
zbMATH Keywords: coordinate
zbMATH Keywords: coordinate polynomial
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
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full work available at URL: s0022-4049(03)00070-7
OpenAlex ID: W4213001234
description: scientific article; zbMATH DE number 1993281