A note on Meisters' and Olech's proof of the global asymptotic stability Jacobian conjecture
type: Item
instance of: scholarly article
title: A note on Meisters' and Olech's proof of the global asymptotic stability Jacobian conjecture
zbMATH Open document ID: 0752.12002
published in: Pacific Journal of Mathematics
publication date: 1992-06-28T00:00:00Z
review text: Let \(k\) be a field of characteristic zero, and let \(F: k^ n\to k^ n\) be a polynomial map. Assume that at every point \(x\in k^ n\), the Jacobian of \(F\) has rank \(n\). Then there exists a positive integer \(N\) such that, for every \(x\in k^ n\), the fiber \(F^{-1}(x)\) has at most \(N\) elements. This theorem was proved by \textit{G. Meisters} and \textit{C. Olech} [Analyse mathématique et applications, Contrib. Honneur J.-L. Lions, 373-381 (1988; Zbl 0668.34048)] for the case \(n=2\) and \(k=\mathbb{R}\) using topological methods and used for proving a result on global stability. The author gives a proof using the theory of \({\mathcal D}\)- modules of differential equations.
Mathematics Subject Classification ID: 12E99
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 32709
zbMATH Keywords: polynomial map
zbMATH Keywords: Jacobian
zbMATH Keywords: global stability
zbMATH Keywords: \({\mathcal D}\)-modules
Wikidata QID: Q123237705
author: Arno van den Essen
reviewed by: Karl-Heinz Kiyek
MaRDI profile type: Publication
full work available at URL: pjm.1991.151.351
OpenAlex ID: W2082644053
DOI: 10.2140/PJM.1991.151.351
description: scientific article; zbMATH DE number 32709