On simple derivations and the group of polynomial automorphisms commuting with certain derivations

In the paper, we first study the subgroup of $K$-automorphisms of $K[x_1,\allowbreak \ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$. We show that the subgroup of $K$-automorphisms of $K[x_1,\ldots,x_n]$ which commutes with a simple derivation of $K[x_1,\ldots,x_n]$ consists of translations under certain hypothesis. We also prove that the subgroup of $K$-automorphisms of $K[x_1,x_2]$ which commutes with a simple derivation is trivial under the same hypothesis, and the subgroup of $K$-automorphisms of $K[x_1,\ldots,x_n]$ which commutes with simple Shamsuddin derivations is trivial under the same hypothesis. Then we give an affirmative answer to the conjecture proposed in \cite{13} for $n=2$.