A resultant criterion and formula for the inversion of a polynomial map in two variables
type: Item
instance of: scholarly article
title: A resultant criterion and formula for the inversion of a polynomial map in two variables
zbMATH Open document ID: 0704.13009
DOI: 10.1016/0022-4049(90)90002-Y
author: Kossivi Adjamagbo
author: Arno van den Essen
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let k be a field of characteristic 0 and consider a map F: \(k^ n\to k^ n\) defined by polynomials \(F_ 1,...,F_ n\in k[X_ 1,...,X_ n]\). The Jacobian conjecture states that F is invertible (and the inverse is a polynomial map) if and only if \(\det (\partial F_ i/\partial x_ j)\in k^*\). In the paper under review the case \(n=2\) is studied. A criterion in terms of resultants for the invertibility of the map is given. Especially there is an equivalent condition to the Jacobian Umkehrproblem: \(\det (\partial F_ i/\partial X_ j)\in k^*\) implies \((d/dX_ i)R_{X_{3-i}}(F_ 1,F_ 2)\in k^*\), \(i=1,2\). Here \(R_{X_ i}\) is the resultant with respect to \(X_ i\).
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 14E99
zbMATH DE Number: 4154588
zbMATH Keywords: Jacobian conjecture
zbMATH Keywords: Jacobian Umkehrproblem
reviewed by: Gerhard Pfister
MaRDI profile type: Publication
cites work: A differential criterion and formula for the inversion of a polynomial map in several variables
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: An inversion formula for two polynomials in two variables
cites work: A criterion to decide if a polynomial map is invertible and to compute the inverse
description: scientific article; zbMATH DE number 4154588