Bifurcation set, M-tameness, Asymptotic critical values and Newton polyhedrons

Abstract

Let $F=(F_1,F_2,...,F_m):{\mathbb{C}}^n\to {\mathbb{C}}^m$ be a polynomial dominant mapping with $n>m$. In this paper we give the relations between the bifurcation set of $F$ and the set of values where $F$ is not M-tame as well as the set of generalized critical values of $F$. We also construct explicitly a proper subset of ${\mathbb{C}}^m$ in terms of the Newton polyhedrons of $F_1,F_2,...,F_m$ and show that it contains the bifurcation set of $F$. In the case $m=n-1$ we show that $F$ is a locally $C^{\infty }$-trivial fibration if and only if it is a locally $C^0$-trivial fibration.

12 pages; accepted for publication in Kodai M. J. Add comment in Remark 2.1, Remark 3.5 and correct Definition 3.2

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