Tame and wild coordinates of K[z][x, y]

Abstract

Let K be a field of characteristic zero. We characterize coordinates and tame coordinates in K[z][x, y], i.e. the images of x respectively under all automorphisms and under the tame automorphisms of K[z][x; y]. We also construct a new large class of wild automorphisms of K[z][x; y] which maps x to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of K[z][x, y, t].