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A note on homogeneous rank $2$ locally nilpotent derivations on $k[X,Y,Z]$

Parnashree Ghosh

Abstract

In this article we show that for every prime number $p$, any irreducible homogeneous locally nilpotent derivations of rank $2$ and degree $p-2$ are triangularizable. Further, we describe the structure of irreducible non-triangularizable homogeneous locally nilpotent derivations of rank $2$ and degree $pq-2$, where $p,q$ are prime numbers. Consequently, we give explicit descriptions of the generators of the image ideals of certain homogeneous locally nilpotent derivations of rank $2$.

A note on homogeneous rank $2$ locally nilpotent derivations on $k[X,Y,Z]$

Abstract

In this article we show that for every prime number , any irreducible homogeneous locally nilpotent derivations of rank and degree are triangularizable. Further, we describe the structure of irreducible non-triangularizable homogeneous locally nilpotent derivations of rank and degree , where are prime numbers. Consequently, we give explicit descriptions of the generators of the image ideals of certain homogeneous locally nilpotent derivations of rank .
Paper Structure (6 sections, 22 theorems, 48 equations)

This paper contains 6 sections, 22 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous locally nilpotent derivations of rank two
  4. An application: Finding generators of image ideals
  5. Definitions and preliminary results
  6. Generators of the image ideals

Key Result

Lemma 2.1

Let $B$ be an integral domain containing $k$, $D$ a non-trivial locally nilpotent derivation on $B$, and $A=ker(D)$. Then the following statements hold:

Theorems & Definitions (39)

  • Definition 2.1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • ...and 29 more