On plane polynomial automorphisms commuting with simple derivations (Q343590)

Let \(\mathbb{K}\) be an algebraically closed field of characteristic zero, \(D:\mathbb{K}[x, y]\rightarrow \mathbb{K}[x, y]\) a derivation of the polynomial ring in two variables over \(\mathbb{K}\), and \(\mathrm{Aut}(D)\) the group of all \(\mathbb{K}\)-linear automorphisms of \(\mathbb{K}[x, y]\) commuting with \(D\). The main results of the paper under review are the following two statements.NEWLINENEWLINETheorem 1. If the derivation \(D\) is simple (that is, \(D\) does not stabilize non-trivial ideals), then the group \(\mathrm{Aut}(D)\) is trivial.NEWLINENEWLINETheorem 2. Let \(D\) be a Shamsuddin derivation, that is, \(D = \partial_{x} + (ay+b)\partial_{y}\) with \(a, b\in \mathbb{K}[x]\). If \(a\neq 0\), then \(D\) is simple if and only if \(\mathrm{Aut}(D)\) is trivial.NEWLINENEWLINEThe authors also obtain some properties of elements of \(\mathrm{Aut}(D)\) (in particular, they show that if \(D\) is a simple derivation and an automorphism \(\rho\in \mathrm{Aut}(D)\) stabilizes the ideal generated by \(x\) in \(\mathbb{K}[x, y]\), then \(\rho\) is an identity mapping) and give some illustrating examples.