Simple derivations and their images

TL;DR

This paper characterizes when certain polynomial derivations are simple and when their images form Mathieu-Zhao spaces, providing conditions based on polynomial degrees and coefficients.

Contribution

It offers new criteria for simplicity of derivations and characterizes Mathieu-Zhao spaces for specific derivations in polynomial rings.

Findings

Derivation $D=yrac{ ext{partial}}{ ext{partial}x} + (a_2(x)y^2 + a_1(x)y + a_0(x))rac{ ext{partial}}{ ext{partial}y}$ is simple iff conditions on $a_i(x)$ hold.

The image of $D=rac{ ext{partial}}{ ext{partial}x} + ext{sum} ext{ of } ext{functions} imes y_i^{k_i}rac{ ext{partial}}{ ext{partial}y_i}$ is a Mathieu-Zhao space iff $D$ is locally finite.

The image of $D= ext{sum} ext{ of } ext{functions} imes y_i^{k_i}rac{ ext{partial}}{ ext{partial}y_i}$ is a Mathieu-Zhao space iff all $k_i eq 1$ for $i=1, ext{to},n$, with $n extgreater 1$.

Abstract

In the paper, we prove that the derivation $D=y\partial_x+(a_2(x)y^2+a_1(x)y+a_0(x))\partial_y$ of $K[x,y]$ with $a_2(x),a_1(x),a_0(x)\in K[x]$ is simple iff the following conditions hold: $(1)$ $a_0(x)\in K^*$, $(2)$ $\deg a_1(x)\geq1$ or $\deg a_2(x)\geq1$, $(3)$ there exist no $l\in K^*$ such that $a_2(x)=la_1(x)-l^2a_0(x)$. In addition, we prove that the image of the derivation $D=\partial_x+{\sum_{i=1}^n \gamma_i(x) y_i^{k_i}}{\partial_i}$ is a Mathieu-Zhao space iff $D$ is locally finite. Moreover, we prove that the image of the derivation $D={\sum_{i=1}^n \gamma_i y_i^{k_i}}{\partial_i}$ of $K[y_1,\ldots,y_n]$ is a Mathieu-Zhao space iff $k_i\leq 1$ for all $1\leq i\leq n$, $n\geq 2$.