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

Parnashree Ghosh

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 $°(D) \leqslant 3$. 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 $°(D) \leqslant 3$.

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

Abstract

Let be a field of characteristic zero and a -algebra. In this paper we study homogeneous -lnds on with respect to the standard weights . We show that when is a PID, can be at most if . As a consequence we obtain a certain class of homogeneous lnds on whose kernel is . Further when is a Dedekind domain, we give a bound for minimum number of generators of as an -algebra if .
Paper Structure (4 sections, 21 theorems, 54 equations)

This paper contains 4 sections, 21 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rank and kernel of homogeneous $R$-lnds on $R[X,Y,Z]$
  4. Appendix

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic zero, and $D$ be a homogeneous locally nilpotent derivation on $k[X,Y,Z] (=k^{[3]})$. If $\deg(D) \leqslant 3$, then $rank(D) \leqslant 2$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 28 more