On the Image Conjecture for Locally Finite Derivations and $\mathcal E$-Derivations

TL;DR

This paper proves several cases of the LFED Conjecture for various algebraic structures, including fields of fractions, formal power series, and Laurent series, and discusses its relation to the Duistermaat-van der Kallen Theorem.

Contribution

It establishes the LFED Conjecture for key algebraic domains and explores its connection with existing theorems, advancing understanding of locally finite derivations.

Findings

LFED Conjecture holds for $k(x)$, $k[[x]]$, and $k[[x]][x^{-1}]$

Relation between LFED Conjecture and Duistermaat-van der Kallen Theorem analyzed

Conjecture verified in several integral domains

Abstract

Some cases of the LFED Conjecture, proposed by the second author [Z3], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal power series algebra $k[[x]][x^{-1}]$, where $x=(x_1, x_2, \dots, x_n)$ denotes $n$ commutative free variables and $k$ a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat-van der Kallen Theorem [DK] is also discussed and emphasized.