Adapting coordinates
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 \(A[y,z]\).
scientific article; zbMATH DE number 1993281