Some results on the Jacobian conjecture in higher dimension
type: Item
instance of: scholarly article
title: Some results on the Jacobian conjecture in higher dimension
zbMATH Open document ID: 0811.13016
DOI: 10.1016/0022-4049(94)90013-2
author: Jeffrey Lang
author: Samer Maslamani
published in: Journal of Pure and Applied Algebra
publication date: 1995-05-01T00:00:00Z
review text: Let \(k\) be a field of characteristic 0 and \(k[x]\) the polynomial ring in \(n\) variables \(x_ 1, \dots, x_ n\) over \(k\), \(F = (F_ 1,\dots, F_ n)\), \(F_ i \in k[x]\), \(i = 1, \dots,n\). The authors state that if \(F_ i = xg_ i\), \(g_ i \in k[x]\) and \(\text{det} (\partial F_ i/ \partial x_ j) \in k^*\), then \(F_ i = \alpha_ i x_ i\) with \(\alpha_ i \in k^*\). They use this result to prove Keller's Jacobian conjecture when each \(F_ i = x_ i + \alpha_ i M_ i\) where \(\alpha_ i \in k\) and \(M_ i\) are monomials. In the proofs of proposition 6 and lemma 7 there are some fragments not clear for the reviewer.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14E07
zbMATH DE Number: 626263
zbMATH Keywords: polynomial automorphisms
zbMATH Keywords: Jacobian conjecture
Wikidata QID: Q123023040
reviewed by: Przemysław Skibiński
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90013-2
OpenAlex ID: W2004927180
cites work: Lectures on expansion techniques in algebraic geometry. With notes by Balwant Singh
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Q3039466
description: scientific article; zbMATH DE number 626263