Length 2 variables of \(A[x,y]\) and transfer (Q2724119)

Let \(A\) be a commutative ring. When \(A\) is a field, the automorphism group of \(A[x_1, \dots , x_n]\) is well known [\textit{M. Nagata}, ``On automorphism group of k(x,y)'', Dep. Math., Kyoto Univ., Lect. Math. 5 (Tokyo 1972; Zbl 0306.14001)]. The authors consider the general case for \(n=2\), when the study of automorphisms reduces to that of variables. They construct a class of length 2 variables, without any assumption on \(A\). Under further hypotheses, more precise results are given: NEWLINENEWLINENEWLINEIf \(A\) is an integral domain, the authors characterize the tame variables in this class; if \(A\) is UFD, all the variables in this class are stably tame. In particular, the Smith variables are among those constructed in the article, if moreover \(\mathbb{Q} \subset A\). Several examples are given. In the final section, these results are used to find variables in \(A[x_1, \dots , x_n]\) by induction on \(n\).