New stably tame automorphisms of polynomial algebras
The paper under review deals with automorphisms of (commutative) polynomial algebras. Suppose \(K[X,Y]\) is the polynomial algebra in \(n\) variables \(X\) and \(m\) variables \(Y\) over a field \(K\) of characteristic 0. The authors consider an appropriate locally nilpotent derivation \(\delta\) of \(K[X,Y]\). It is defined as \(\delta(y_i)=0\) for all \(i\), and it is a \(K[Y]\)-affine transformation over the free \(K[Y]\)-module with free generators \(x_1,\dots,x_n\). The main result of the paper states that for \(w\in\ker(\delta)\) the automorphism \(\exp(w\delta)\) is stably tame. (Recall that an automorphism \(\phi\) is stably tame for \(K[X]\) if its extension to \(K[X,X_1]\) for some finite set of variables \(X_1\) is tame. Here the extension of \(\phi\) acts as \(\phi\) on \(K[X]\) and fixes the variables in \(X_1\).) Further the authors give an algorithm for constructing stably tame automorphisms, and investigate the stable tameness of some known automorphisms.
scientific article; zbMATH DE number 1441827