A counterexample to a conjecture of Wright on homogeneous polynomial maps associated with rooted trees
type: Item
instance of: scholarly article
title: A counterexample to a conjecture of Wright on homogeneous polynomial maps associated with rooted trees
zbMATH Open document ID: 1019.14024
DOI: 10.1016/S0022-4049(02)00332-8
author: Piotr Ossowski
published in: Journal of Pure and Applied Algebra
publication date: 2003-06-18T00:00:00Z
review text: Let \(K\) be a field and \(F_1,\dots, F_n\in K[x]\), \(X=(X_1, \dots, X_n)\), \(F=(F_1, \dots, F_n)\). Let \(J(F)=({\partial F_i\over \partial X_j})\) be the Jacobian matrix and assume that \(\det(J(F))\in K\subset\{0\}\). The Jacobian conjecture states that there exists a polynomial map \(G:K^n\to K^n\) such that \(G(F)=X\). There is a conjecture of \textit{D. Wright} [J. Pure Appl. Algebra 48, 199--219 (1987; Zbl 0666.12017)] concerning the case \(F=X-H\), \(H\) homogeneous of degree \(d\geq 3\) and the Jacobian matrix \(J(H)\) being nilpotent, which implies for \(d=3\) the Jacobian conjecture. It is proven that Wright's conjecture is false for \(d\geq 3\) and \(n\geq 5\).
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 1930185
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: Wright's conjecture
MaRDI profile type: Publication
Wikidata QID: Q122930629
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description: scientific article; zbMATH DE number 1930185