The Jacobian conjecture in case of rank or corank less than three
type: Item
instance of: scholarly article
title: The Jacobian conjecture in case of rank or corank less than three
zbMATH Open document ID: 0781.13015
DOI: 10.1016/0022-4049(93)90140-O
author: Ludwik M. Drużkowski
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: The unproved Jacobian conjecture is the following: if \(F=(F_ 1,\ldots,F_ n):\mathbb{C}^ n\mapsto\mathbb{C}^ n\) is a polynomial mapping with nonzero constant jacobian, then \(F\) is a polynomial automorphism of \(\mathbb{C}^ n\). It was shown by \textit{L. M. Drużkowski} [Math. Ann. 264, 303-313 (1983; Zbl 0504.13006)] that it is sufficient to prove the Jacobian conjecture for every \(n>1\) only for \(F\) having the following form \(F_ j(x_ 1,\ldots,x_ n)=x_ j+(a^ 1_ j+\cdots+a^ n_ jx_ n)^ 3\), \(a^ i_ j\in\mathbb{C}\). The author proves that the Jacobian conjecture holds for \(F\) of the above form in the cases: \(\text{rank}[a^ i_ j]=1, 2\) and \(\text{corank}[a^ i_ j]=1, 2\). Actually, he proves a little more. Namely, in these cases the mapping \(F-F(0)\) is a tame automorphism i.e. a finite composition of linear automorphisms and shears. The latter are polynomial automorphisms of \(\mathbb{C}^ n\) of the form \(T(x_ 1,\ldots,x_ n)=(x_ 1,\ldots,x_{i-1},x_ i+f(x_ 1,\ldots,x_{i- 1},x_{i+1},\ldots, x_ n),x_{i+1},\ldots,x_ n)\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14H37
zbMATH DE Number: 167298
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: polynomial automorphism
Wikidata QID: Q122942078
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
cites work: Volume-preserving automorphisms of C⊃n⊃
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
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cites work: Stably tame automorphisms
cites work: The jacobian conjecture: linear triangularizatlon for cubics in dimension three
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description: scientific article; zbMATH DE number 167298