Nilpotent Jacobians in dimension three
type: Item
instance of: scholarly article
title: Nilpotent Jacobians in dimension three
zbMATH Open document ID: 1085.14052
author: Marc Chamberland
author: Arno van den Essen
published in: Journal of Pure and Applied Algebra
publication date: 2006-02-07T00:00:00Z
full work available at URL: http://hdl.handle.net/2066/35575
review text: It is known that if \(F=x+H:k^{n}\rightarrow k^{n}\) (\(k\) - a field of characteristic zero) is a polynomial Keller's mapping (i.e. \(\text{Jac}~F= \text{const}.\neq 0\)) then the Jacobian matrix \(JH\) of \(H\) is nilpotent. It is known that to solve the Jacobian conjecture it suffices to study the case of homogeneous \(H.\) This, in turn, is related to the homogeneous dependence problem (which is still open): if \(H\) is homogeneous with \(JH\) nilpotent and \(H(0)=0,\) then \(H_{1},\ldots ,H_{n}\) are linearly dependent. In the inhomogeneous case the answer to the above problem is negative due to a counterexample given by the second author. The paper under review studies the polynomial mappings \(H\) with \(JH\) nilpotent in dimension \(n=3.\) The authors completely classify such mappings of the form \(H=(u(x,y),v(x,y),h(u(x,y),v(x,y)))\) with \(JH\) nilpotent for which components are linear independent.
Mathematics Subject Classification ID: 14R15
zbMATH DE Number: 5004397
zbMATH Keywords: nilpotent matrix
zbMATH Keywords: polynomial mapping
zbMATH Keywords: homogeneous dependence problem
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
OpenAlex ID: W2121921306
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DOI: 10.1016/J.JPAA.2005.06.003
description: scientific article; zbMATH DE number 5004397