The Jacobian conjecture for symmetric Jacobian matrices.
type: Item
instance of: scholarly article
title: The Jacobian conjecture for symmetric Jacobian matrices.
zbMATH Open document ID: 1050.14059
author: Sherwood Washburn
author: Arno van den Essen
published in: Journal of Pure and Applied Algebra
publication date: 2004-07-12T00:00:00Z
review text: Let \(F = (F_1,\ldots,F_n) : \mathbb C^n \longrightarrow \mathbb C^n\) be a polynomial map and \(J(F) = (\partial F_i/\partial x_j)\) the Jacobian matrix of \(F\). The Jacobian conjecture asserts that \(F\) is invertible if det\((J(F)) \in \mathbb C^*\). It is known that it suffices to prove the conjecture for all \(n \geq 2\) and all polynomial mappings of the form \(F = x + H, H=(H_1,\ldots,H_n)\) and \(H_i\) is either zero or homogeneous of degree 3. It is proved that for all \(n \leq 4\) the Jacobian conjecture holds for all polynomial mappings \(F = x + H\) where \(H\) is homogeneous of degree \(> 1\) and \(J(F)\) is symmetric. The same assumptions for \(F : \mathbb R^n \longrightarrow \mathbb R^n\) imply \(F = x\).
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14R15
zbMATH DE Number: 2078854
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: polynomial mappings
MaRDI profile type: Publication
Wikidata QID: Q122910914
full work available at URL: j.jpaa.2003.10.020
OpenAlex ID: W1999171030
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DOI: 10.1016/J.JPAA.2003.10.020
description: scientific article; zbMATH DE number 2078854