On bifurcation points of a complex polynomial

Jelonek, Zbigniew

doi:10.1090/S0002-9939-02-06822-3

On bifurcation points of a complex polynomial
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by Zbigniew Jelonek

Proc. Amer. Math. Soc. 131 (2003), 1361-1367

DOI: https://doi.org/10.1090/S0002-9939-02-06822-3

Published electronically: December 16, 2002

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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, $\#\tilde {K}_\infty (f)\le (d-1)^{n-1}.$ We also compute the set $\tilde {K} (f)$ effectively.

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