The LFED Conjecture for some $\mathcal{E}$-derivations

TL;DR

This paper investigates the LFED Conjecture for certain $ ext{E}$-derivations over polynomial rings, proving it in specific cases including affine polynomial homomorphisms and some three-variable instances.

Contribution

It establishes the LFED Conjecture for all $ ext{E}$-derivations of the form $I - ext{affine polynomial homomorphism}$ in two variables and some cases in three variables.

Findings

Im $ ext{delta}$ is a Mathieu-Zhao space in some cases

LFED Conjecture holds for $ ext{delta} = I - ext{affine polynomial}$ in two variables

LFED Conjecture is true for some $ ext{delta}$ in three variables

Abstract

Let $K$ be an algebraically closed field of characteristic zero, $\delta$ a nonzero $\mathcal{E}$-derivation of $K[x]$. We first prove that $\operatorname{Im}\delta$ is a Mathieu-Zhao space of $K[x]$ in some cases. Then we prove that LFED Conjecture is true for all $\delta=I-\phi$, where $\phi$ is an affine polynomial homomorphism of $K[x_1,x_2]$. Finally, we prove that LFED Conjecture is true for some $\delta$ of $K[x_1,x_2,x_3]$.