Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$

TL;DR

This paper investigates homogeneous locally nilpotent derivations on polynomial rings over various rings, establishing bounds on their rank and generators, especially over PIDs and Dedekind domains, with implications for the structure of their kernels.

Contribution

It provides new bounds on the rank and generators of homogeneous locally nilpotent derivations on polynomial rings over PIDs and Dedekind domains, extending understanding of their algebraic structure.

Findings

Rank of derivations is at most 2 for degree ≤ 3 over PIDs.

Identifies a class of homogeneous locally nilpotent derivations with kernels isomorphic to polynomial rings.

Provides bounds on the number of generators of the kernel over Dedekind domains.

Abstract

Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In this paper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to the standard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be at most $2$ if $\deg(D) \leqslant 3$. As a consequence we obtain a certain class of homogeneous lnds on $k^{[4]}$ whose kernel is $k^{[3]}$. Further when $R$ is a Dedekind domain, we give a bound for minimum number of generators of $\ker(D)$ as an $R$-algebra if $\deg(D) \leqslant 3$.