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On topological approaches to the Jacobian conjecture in ℂn
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 63 / Issue 3 / August 2020
- Published online by Cambridge University Press:
- 07 May 2020, pp. 666-675
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- Article
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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.
