Generic Coordinate Systems in Two Variables Over a Principal Ideal Domain

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