Quadratic linear Keller maps of nilpotency index three
type: Item
instance of: scholarly article
title: Quadratic linear Keller maps of nilpotency index three
zbMATH Open document ID: 1147.14031
author: Xiankun Du
author: Xiaosong Sun
author: Dayan Liu
published in: Linear Algebra and its Applications
publication date: 2008-06-09T00:00:00Z
review text: Let \(f=x+H:\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}\) be a polynomial map such that \(H\) is homogeneous of degree \(d\geq 2.\) Let \(F_{A}=y+(Ay)^{d}: \mathbb{C}^{n}\rightarrow \mathbb{C}^{n},\) \(A\in M_{n,n}(\mathbb{C}),\) be a power linear map such that \(\;f\) and \(F_{A}\) are a generalized Gorni-Zampieri pair i.e. there exist matrices \(B\in M_{n,N}(\mathbb{C})\) and \(C\in M_{N,n}(\mathbb{C})\) such that 1. \(f(x)=BF_{A}(C(x)),\) 2. \(BC=I_{n},\) 3. \(\ker B=\ker A.\) For a given \(f\) such a \(F_{A}\) always exists. The authors prove the following relations between \(f\) and \(F_{A}:\) 1. for indices of nipotency \[ \operatorname{Ind}\operatorname{Jac}(A(y))^{d}=\operatorname{Ind}\operatorname{Jac}H+1, \] 2. \(f\) is linearly triangularizable if and only if \(F_{A}\) is linearly triangularizable. As a consequence they obtain that quadratic linear Keller map \( F_{A}=y+(Ay)^{2}\) with nilpotency index three i.e. \(\left( \operatorname{Jac} (A(y))^{2}\right) ^{3}=0,\) is linearly triangularizable.
Mathematics Subject Classification ID: 14R15
zbMATH DE Number: 5285754
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: linearly triangularizable
zbMATH Keywords: power linear map
zbMATH Keywords: Gorni-Zampieri map
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
full work available at URL: j.laa.2008.01.032
OpenAlex ID: W1966731594
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Cubic linear Keller maps
cites work: Quadratic linear Keller maps
cites work: Power Linear Keller Maps of Nilpotency Index Two
cites work: Power linear Keller maps of rank two are linearly triangularizable
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: On cubic-linear polynomial mappings
cites work: The linear dependence problem for power linear maps
cites work: Strong nilpotence holds in dimensions up to five only∗
cites work: Poisson brackets and two-generated subalgebras of rings of polynomials
cites work: The tame and the wild automorphisms of polynomial rings in three variables
cites work: Power linear Keller maps with ditto triangularizations
cites work: Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: A Jacobian criterion for separability
DOI: 10.1016/J.LAA.2008.01.032
description: scientific article; zbMATH DE number 5285754