Images of locally finite $\mathcal{E}$-derivations of bivariate polynomial algebras

Abstract

This paper presents an $\mathcal{E}$-derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite $K$-$\mathcal{E}$-derivation of polynomial algebras in two variables over a field $K$ of characteristic zero is a Mathieu-Zhao subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables.