On polynomial automorphisms commuting with a simple derivation

Abstract

Let D be a simple derivation of the polynomial ring k[x1,…,xn], where k is an algebraically closed field of characteristic zero, and denote by Aut(D)⊂Aut(k[x1,…,xn]) the subgroup of k-automorphisms commuting with D. We show that the connected component of Aut(D) passing through the identity is a unipotent algebraic group of dimension at most n−2, this bound being sharp. Moreover, Aut(D) is an algebraic group if and only if it is a connected ind-group. Given a simple derivation D, we characterize when Aut(D) contains a normal subgroup of translations. As an application of our techniques we show that if n=3, then either Aut(D) is a discrete group or it is isomorphic to the additive group acting by translations, and give some insight on the case n=4.