On the density of image of differential operators generated by polynomials
type: Item
instance of: scholarly article
title: On the density of image of differential operators generated by polynomials
zbMATH Open document ID: 0682.13002
DOI: 10.1007/BF02820481
author: Yosef Stein
published in: Journal d'Analyse Mathématique
publication date: 1989-01-01T00:00:00Z
review text: Let P(x,y) be a polynomial in two variables (over the complex numbers, say) and define the partial differential operator \[ D_ P(x,y)=\frac{\partial P}{\partial x}\frac{\partial f}{\partial y}- \frac{\partial P}{\partial y}\frac{\partial f}{\partial x}. \] The author proves that the image of \(D_ P\) is dense in the ring of entire functions on \({\mathbb{C}}^ 2\) if and only if P is a generator for the ring \({\mathbb{C}}[x,y]\). - Obviously the result is related to the Jacobian conjecture, and also has intrinsic interested. The proof proceeds by a careful analysis of critical points of a polynomial of two variables.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 54C40
zbMATH DE Number: 4116674
zbMATH Keywords: polynomial ring
zbMATH Keywords: Jacobian conjecture
reviewed by: Steven G. Krantz
MaRDI profile type: Publication
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Sur les algèbres de Weyl
cites work: On linear differential operators related to the Jacobian conjecture
full work available at URL: https://doi.org/10.1007/bf02820481
OpenAlex ID: W2083778649
description: scientific article; zbMATH DE number 4116674