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

Abstract

Let F = (F₁, F₂, ..., F_m): Cⁿ → 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 C^m in terms of the Newton polyhedrons of F₁, F₂, ..., 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^∞-trivial fibration if and only if it is a locally C⁰-trivial fibration.