Power linear Keller maps of dimension four
type: Item
instance of: scholarly article
title: Power linear Keller maps of dimension four
zbMATH Open document ID: 1009.14011
DOI: 10.1016/S0022-4049(01)00065-2
author: Charles Ching-An Cheng
published in: Journal of Pure and Applied Algebra
publication date: 2002-06-24T00:00:00Z
review text: The author proves the following fact connected to the Jacobian conjecture and tame generators conjecture. Let \(K\) be a field of characteristic zero. A polynomial map \(F=(F_1, \dots,F_n): K^n\to K^n\) is called power linear iff \(F_i(X_1,\dots,X_n) =X_i+(A_i(X_1,\dots,X_n))^{d}\) where \(A_i\) are linear forms and \(d>1\). The main theorem is: If \(n=4\) and \(\text{Jac}F\in K^*\) then the above \(F\) is linearly triangularizable, i.e. there exists a linear invertible map \(\phi\) such that \(\phi\cdot F\cdot\phi^{-1}(X_1,\dots,X_n)=(X_1+p_1, \dots,X_n+p_n),\quad p_i\in K[X_{i+1},\dots,X_n].\) Hence \(F\) (as above) is tame and invertible. The author [\textit{C. C.-A. Cheng}, Proc. Am. Math. Soc. 129, No. 10, 2819-2822 (2001; Zbl 0991.14022)] proved the same result for \(n=3\) but under a slight general definition of power linear maps.
reviewed by: Tadeusz Krasiński
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 1758340
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: power linear map
zbMATH Keywords: tame generators conjecture
zbMATH Keywords: linearly triangularizable polynomial map
MaRDI profile type: Publication
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Power linear Keller maps of dimension three
cites work: Cubic linear Keller maps
cites work: An effective approach to Keller's Jacobian conjecture
cites work: The Jacobian conjecture in case of rank or corank less than three
cites work: Q4848530
cites work: Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture
cites work: Über ganze birationale Transformationen der Ebene.
cites work: Q5818521
cites work: Strong nilpotence holds in dimensions up to five only∗
cites work: The jacobian conjecture: linear triangularizatlon for cubics in dimension three
description: scientific article; zbMATH DE number 1758340