On bifurcation points of a complex polynomial期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On bifurcation points of a complex polynomial

Authors:	Zbigniew Jelonek

Institution:	Instytut Matematyczny, Polska Akademia Nauk, Sw. Tomasza 30, 31-027 Kraków, Poland

Abstract:	Let $f: \mathbb{C} ^n \to \mathbb{C}$ be a polynomial of degree . 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 (hence in particular the set of bifurcation points of ) has at most points. Moreover, $\char93 \tilde{K}_\infty (f)\le (d-1)^{n-1}.$ We also compute the set $\tilde{K} (f)$ effectively.

Keywords:	Polynomial mapping fibration bifurcation points the set of points over which a polynomial mapping is not proper

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