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On topological approaches to the Jacobian conjecture in ℂ n

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...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 2020-08, Vol.63 (3), p.666-675
Main Authors: Braun, Francisco, Dias, Luis Renato Gonçalves, Venato-Santos, Jean
Format: Article
Language:English
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Summary: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.
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091520000061