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.