Stably co-tame polynomial automorphisms over commutative rings (Q1697191)

Let \(R\) be a commutative associative ring with 1 and let \(R[\mathbf x]=R[x_1,\ldots,x_n]\) be the polynomial ring in \(n\) variables over \(R\). Let \(\text{GA}_n(R)=\text{Aut}_RR[\mathbf x]\) be the group of automorphisms of the \(R\)-algebra \(R[\mathbf x]\) and let \(\text{T}_n(R)\) be the subgroup of the tame automorphisms (generated by the affine and the triangular automorphisms). Following [\textit{E. Edo}, Commun. Algebra 41, No. 12, 4694--4710 (2013; Zbl 1284.14085)], an automorphism \(\phi\) of \(R[\mathbf x]\) is called co-tame, if the group generated by \(\phi\) and the affine automorphisms contains all tame automorphisms. In the paper under review the author considers the larger class of stably co-tame automorphisms. Assuming that the automorphisms of \(\text{GA}_n(R)\) fix the variable \(x_{n+1}\), one embeds \(\text{GA}_n(R)\) into \(\text{GA}_{n+1}(R)\). Then \(\phi\in\text{GA}_n(R)\) is stably co-tame if the subgroup of \(\text{GA}_{n+1}(R)\) generated by \(\phi\) and the affine automorphisms in \(n+1\) variables contains \(\text{T}_n(R)\). The author gives several sufficient conditions for the stable co-tameness of an automorphism \(\phi\). The conditions are also necessary when \(R\) is a field. In the special case of a field of characteristic 0 an automorphism is stably co-tame if and only if it is not affine.