A counterexample to a conjecture related to the Jacobian problem
type: Item
instance of: scholarly article
title: A counterexample to a conjecture related to the Jacobian problem
zbMATH Open document ID: 0868.58012
DOI: 10.1007/BF02304778
published in: Mathematical Notes
publication date: 1997-08-21T00:00:00Z
review text: The author gives a counterexample to a conjecture, posed by Vitushkin, connected with the Jacobian conjecture in \(\mathbb{R}^2\). Namely, he constructs a two-dimensional manifold \(M\) comprising one open two-dimensional cell \(N\) (homeomorphic to \(\mathbb{R}^2\)) and three open one-dimensional cells \(R\) (homeomorphic to \(\mathbb{R}\)) and a mapping \(F:M\to\mathbb{R}^2\) such that: 1. \(F\) is locally homeomorphic on \(N\); 2. in a neighbourhood of each \(P\in R\) there exists a coordinate system in which \(F\) is given by \(F(x,y)=(x,y^k)\) with \(k=1\) or \(k=2\); 3. \(F\) maps each \(R\) into a curve in \(\mathbb{R}^2\) that tends to infinity along the same asymptotic direction as a point of \(R\) tends to the ends of \(R\).
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 898948
zbMATH Keywords: counterexample
zbMATH Keywords: Jacobian conjecture
Wikidata QID: Q123271084
author: F. Dell'Accio
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
cites work: Ganze Cremona-Transformationen
cites work: Q4067204
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: A counterexample to the strong real Jacobian conjecture
full work available at URL: https://doi.org/10.1007/bf02304778
OpenAlex ID: W126242048
description: scientific article; zbMATH DE number 898948