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

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.

arXiv commentSome minor changes. The final version. Latex 13 pages

References in corpus (4)

• Images of Ideals under Derivations and $\mathcal{E}$-Derivations of Univariate Polynomial Algebras over a Field of Characteristic Zero
• The LFED and LNED Conjectures for Laurent Polynomial Algebras
• Some Open Problems on Locally Finite or Locally Nilpotent Derivations and $\mathcal{E}$-Derivations
• Idempotents in Intersection of the Kernel and the Image of Locally Finite Derivations and $\mathcal{E}$-derivations