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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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Published in: | Proceedings of the Edinburgh Mathematical Society 2020-08, Vol.63 (3), p.666-675 |
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Main Authors: | , , |
Format: | Article |
Language: | English |
Citations: | Items that this one cites |
Online Access: | Get full text |
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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. |
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ISSN: | 0013-0915 1464-3839 |
DOI: | 10.1017/S0013091520000061 |