Simple rational polynomials and the Jacobian conjecture
type: Item
instance of: scholarly article
title: Simple rational polynomials and the Jacobian conjecture
zbMATH Open document ID: 1152.14050
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2008-08-12T00:00:00Z
review text: The paper under review studies the jacobian conjecture in dimension two, the question as to whether a polynomial function \(F=(f,g): \mathbb C^{2} \to \mathbb C^{2}\) with nonzero constant jacobian \(J(F)\) must be an isomorphism. The author focuses on one of the coordinate functions, say \(f: \mathbb C^{2} \to \mathbb C\). Homogenizing \(f\) yields a rational map \(\mathbb P^{2} \to \mathbb P^{1}\), and after a blow-up \(\pi: Z \to \mathbb P^{2}\) of points of indeterminacy one arrives at a morphism \(\varphi: Z \to \mathbb P^{1}\). The divisor \(\mathcal D = \pi^{-1}(\mathbb P^{2}- \mathbb C^{2})\) at infinity may have several components. A component \(V \subset \mathcal D\) is dicritical if the restriction \(\varphi|_{V}\) is non-constant and \(f\) is called \textit{simple} if the degree of each such restriction is one. The main theorem of the paper says that if \(f\) is simple, the generic fibre of \(f\) is rational, and \(f\) is not a locally trivial fibration, then \(f\) cannot belong to a Jacobian pair. In particular, the Jacobian conjecture holds for maps \(F=(f,g)\) in which \(f\) is simple with generically rational fibres.
reviewed by: Scott Nollet
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 5310673
author: Dũng Tráng Lê
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1210167339
OpenAlex ID: W1618709518
Wikidata QID: Q123310056
cites work: Q5749404
cites work: Q4486513
cites work: Q4195061
cites work: Q4143433
cites work: A geometrical approach to the Jacobian conjecture for \(n=2\)
cites work: Q4854766
cites work: Q3485178
cites work: Rational polynomials of simple type.
DOI: 10.2977/PRIMS/1210167339
description: scientific article; zbMATH DE number 5310673