The centralizer of a locally nilpotent R-derivation of the polynomial R-algebra in two variables

Abstract

Let R be an integral domain containing $Q$ and ξ be an irreducible nontrivial locally nilpotent R-derivation of the polynomial R-algebra A in two variables. In this paper we investigate the group ${\mathrm{Aut}}_R(A,\xi )$ of R-automorphisms of A which commute with ξ. In the case R is a unique factorization domain and the plinth ideal of ξ is principal we give a complete description of the subgroup ${\mathrm{SAut}}_R(A,\xi )$ of ${\mathrm{Aut}}_R(A,\xi )$ consisting of Jacobian one automorphisms. If moreover R contains a field K such that the group of units of R is $K^{\star }$ we prove that ${\mathrm{Aut}}_R(A,\xi )={\mathrm{SAut}}_R(A,\xi )$.