Generic Coordinate Systems in Two Variables Over a Principal Ideal Domain

Abstract

Let \(R\) be a principal ideal domain. In this paper we investigate generic coordinate systems of the polynomial \(R\)-algebra \(A=R^{[2]}\). As an application we prove that for every locally nilpotent \(R\)-derivation \(\xi \) of \(A\) the automorphism \(\exp (\xi )\) is 1-stably tame in an appropriate coordinate system of \(A\). This shows that the well-known result due to Smith, asserting that the Nagata automorphism is 1-stably tame, actually holds in full generality.