Locally nilpotent derivations of polynomial rings
type: Item
instance of: scholarly article
title: Locally nilpotent derivations of polynomial rings
zbMATH Open document ID: 0857.13006
DOI: 10.1007/BF02304113
published in: Mathematical Notes
publication date: 1997-03-11T00:00:00Z
review text: This note is concerned with locally nilpotent derivations, that are derivations \(D\) on a ring \(R\) fulfilling \(\text{Nil} (D)=R\), \(\text{Nil} (D)\) being the set \(\{x\in R; \exists n\in \mathbb{N}\) with \(D^n(x)=0\}\). In particular the following theorem is shown: Let \(F\) be an irreducible algebraic curve in \(\mathbb{A}^2\) given by the equation \(f(x,y)=0\). Then \(F\cong \mathbb{A}^1\) if and only if \(D_f=f_y {\partial \over \partial x} - f_x{\partial \over \partial y}\) is a locally nilpotent derivation of the ring \(k[x,y]\); here \(\mathbb{A}^n\) denotes the \(n\)-dimensional affine space over an algebraically closed field \(k\) with \(\text{char} k=0\). Also the rectifiability of a surface \(G\subset \mathbb{A}^3\) is expressed in terms of locally nilpotent derivations.
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13N99
zbMATH DE Number: 898933
zbMATH Keywords: locally nilpotent derivations
zbMATH Keywords: affine space
author: F. Pakovich
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
cites work: Lectures on the fourteenth problem of Hilbert
cites work: On Zariski problem
full work available at URL: https://doi.org/10.1007/bf02304113
OpenAlex ID: W2025835843
description: scientific article; zbMATH DE number 898933