Images of derivations of polynomial algebras with divergence zero (Q2408822)

Let \(k\) be a field and let \(D: k[x,y] \to k[x,y]\) be a \(k\)-derivation. The \textit{divergence} of \(D\) is \(\text{div} D = \partial_x (D(x)) + \partial_y (D(y))\). According to \textit{W. Zhao} [J. Pure Appl. Algebra 214, 1200--1216 (2010; Zbl 1205.33017)], \(M \subset k[x,y]\) is a \textit{Mathieu} subspace if \(f^m \in M\) for all \(m \geq 1\) implies that for each \(g \in k[x,y]\), \(g f^r \in M\) for all sufficiently large \(r\). In their work on the Jacobian conjecture, \textit{A. van den Essen} et al. ask [J. Pure Appl. Algebra 215, 2130--2134 (2011; Zbl 1229.13022)] whether the image of every \(k\)-derivation \(D\) with \(\text{div} D = 0\) is a Mathieu subspace of \(k[x,y]\). The author gives a negative answer to this question by showing that for a derivation of the form \(D = b x^a y^{b-1} \partial_x - a x^{a-1} y^b \partial_y\) with \(a \geq 1\) and \(b \geq 1\), the image of \(D\) is a Mathieu subspace if and only if \(a=b\).