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

TL;DR

This paper proves that the images of locally finite $\\mathcal{E}$-derivations in bivariate polynomial algebras over characteristic zero fields are Mathieu-Zhao subspaces, confirming the LFED conjecture in this case.

Contribution

It introduces an $\\mathcal{E}$-derivation analogue of a known derivation result and proves the LFED conjecture for two-variable polynomial algebras.

Findings

Images of locally finite $\mathcal{E}$-derivations are Mathieu-Zhao subspaces.

Confirms the LFED conjecture for two-variable polynomial algebras.

Extends derivation results to $\\mathcal{E}$-derivations in polynomial algebra context.

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.