Images of Locally Finite Derivations of Polynomial Algebras in Two Variables

TL;DR

This paper proves that the images of locally finite derivations in two-variable polynomial algebras are Mathieu subspaces and relates this to the Jacobian conjecture, offering new insights into algebraic structure and conjecture equivalences.

Contribution

It establishes that images of locally finite derivations are Mathieu subspaces and links the Jacobian conjecture to Mathieu subspaces in two-variable polynomial algebras.

Findings

Images of locally finite derivations are Mathieu subspaces.

The Jacobian conjecture is equivalent to a property of derivation images.

Provides new algebraic characterizations related to the Jacobian conjecture.

Abstract

In this paper we show that the image of any locally finite $k$-derivation of the polynomial algebra $k[x, y]$ in two variables over a field $k$ of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image $Im D$ of every $k$-derivation $D$ of $k[x, y]$ such that $1\in Im D$ and $div D=0$ is a Mathieu subspace of $k[x, y]$.