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On topological approaches to the Jacobian conjecture in ℂn

Part of: Affine geometry Partial differential equations Differential topology Theory of singularities and catastrophe theory General first-order equations and systems Families, fibrations

Published online by Cambridge University Press:  07 May 2020

Francisco Braun ,
Luis Renato Gonçalves Dias  and
Jean Venato-Santos
Francisco Braun
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos, São Paulo, Brazil ([email protected])
Luis Renato Gonçalves Dias
Affiliation:
Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100 Uberlândia, Minas Gerais, Brazil ([email protected] and [email protected])
Jean Venato-Santos
Affiliation:
Faculdade de Matemática, Universidade Federal de Uberlândia, 38408-100 Uberlândia, Minas Gerais, Brazil ([email protected] and [email protected])
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Abstract

We obtain a new theorem for the non-properness set $S_f$ of a non-singular polynomial mapping $f:\mathbb C^n \to \mathbb C^n$. In particular, our result shows that if f is a counterexample to the Jacobian conjecture, then $S_f\cap Z \neq \emptyset $, for every hypersurface Z dominated by $\mathbb C^{n-1}$ on which some non-singular polynomial $h: \mathbb C^{n}\to \mathbb C$ is constant. Also, we present topological approaches to the Jacobian conjecture in $\mathbb C^n$. As applications, we extend bidimensional results of Rabier, Lê and Weber to higher dimensions.

Keywords

Jacobian conjectureglobal injectivitylocally trivial fibrationsnon-properness set

MSC classification

Primary: 14R15: Jacobian problem 14D06: Fibrations, degenerations 58K15: Topological properties of mappings 57R45: Singularities of differentiable mappings
Secondary: 35F05: Linear first-order equations 35A30: Geometric theory, characteristics, transformations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 666 - 675
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Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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