Images of Ideals under Derivations and $\mathcal{E}$-Derivations of Univariate Polynomial Algebras over a Field of Characteristic Zero

Abstract

Let $K$ be a field of characteristic zero and $x$ a free variable. A $K$-$\mathcal{E}$-derivation of $K[x]$ is a $K$-linear map of the form $I-\varphi$ for some $K$-algebra endomorphism $\varphi$ of $K[x]$, where $I$ denotes the identity map of $K[x]$. In this paper we study the image of an ideal of $K[x]$ under some $K$-derivations and $K$-$\mathcal{E}$-derivations of $K[x]$. We show that the LFED conjecture proposed in [Z4] holds for all $K$-$\mathcal{E}$-derivations and all locally finite $K$-derivations of $K[x]$. We also show that the LNED conjecture proposed in [Z4] holds for all locally nilpotent $K$-derivations of $K[x]$, and also for all locally nilpotent $K$-$\mathcal{E}$-derivations of $K[x]$ and the ideals $uK[x]$ such that either $u=0$, or $\mathrm{deg}u\le 1$, or $u$ has at least one repeated root in the algebraic closure of $K$. As a bi-product, the homogeneous Mathieu subspaces (Mathieu-Zhao spaces) of the univariate polynomial algebra over an arbitrary field have also been classified.

arXiv comment21 pages

References in corpus (3)

• The LFED and LNED Conjectures for Laurent Polynomial Algebras
• The Gaussian Moments Conjecture and the Jacobian Conjecture
• Mathieu Subspaces of Univariate Polynomial Algebras

Cited by in corpus (6)

• 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
• Images of Linear Derivations and Linear E-derivations of K[x 1 ,x 2 ,x 3 ]
• On the Image Conjecture for Locally Finite Derivations and $\mathcal{E}$-Derivations
• The Radical of the Kernel of a Certain Differential Operator and Applications to Locally Algebraic Derivations