Images of Linear Derivations and Linear E-derivations of K[x 1 ,x 2 ,x 3 ]

TL;DR

This paper proves that the images of linear derivations and E-derivations of the polynomial ring in three variables over a field of characteristic zero are Mathieu-Zhao subspaces, confirming the LFED conjecture in this context.

Contribution

It establishes that images of linear K-derivations and K-E-derivations of K[x_1,x_2,x_3] are Mathieu-Zhao subspaces, affirming the LFED conjecture for these cases.

Findings

Images are Mathieu-Zhao subspaces.

Confirms LFED conjecture for linear derivations.

Validates the structure of derivation images in three-variable polynomial rings.

Abstract

Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED conjecture for linear K-derivations and linear K-E-derivations of K[x 1 ,x 2 ,x 3 ].