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On topological approaches to the Jacobian conjecture in ℂn

Published online by Cambridge University Press:  07 May 2020

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])

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.

Type
Research Article
Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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