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On bifurcation points of a complex polynomial

Author: Zbigniew Jelonek
Journal: Proc. Amer. Math. Soc. 131 (2003), 1361-1367
MSC (2000): Primary 14D06, 14Q20, 14R25
Published electronically: December 16, 2002
MathSciNet review: 1949865
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Abstract: Let $f: \mathbb{C} ^n \to \mathbb{C} $ be a polynomial of degree $d$. Assume that the set $\tilde{K}_\infty (f)=\{ y \in \mathbb{C} :$ there is a sequence $x_l\rightarrow\infty $ s.t. $f(x_l)\rightarrow y $ and $\Vert d f(x_l)\Vert\rightarrow 0\}$ is finite. We prove that the set $\tilde{K} (f)= K_0(f)\cup \tilde{K}_\infty (f)$ of generalized critical values of $f$ (hence in particular the set of bifurcation points of $f$) has at most $(d-1)^n$points. Moreover, $\char93 \tilde{K}_\infty (f)\le (d-1)^{n-1}.$ We also compute the set $\tilde{K} (f)$ effectively.

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Additional Information

Zbigniew Jelonek
Affiliation: Instytut Matematyczny, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland
Address at time of publication: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

Keywords: Polynomial mapping, fibration, bifurcation points, the set of points over which a polynomial mapping is not proper
Received by editor(s): April 17, 2001
Received by editor(s) in revised form: January 8, 2002
Published electronically: December 16, 2002
Additional Notes: The author was partially supported by KBN grant number 2P03A01722
Communicated by: Michael Stillman
Article copyright: © Copyright 2002 American Mathematical Society