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

TL;DR

This paper explores the relationship between bifurcation sets, M-tameness, and critical values of polynomial mappings, providing explicit constructions using Newton polyhedrons and analyzing triviality conditions.

Contribution

It establishes connections between bifurcation sets, M-tameness, and critical values, and constructs explicit Newton polyhedron-based subsets containing bifurcation sets.

Findings

Constructs a subset of a0a0 containing the bifurcation set.

Shows the equivalence of smooth and continuous triviality for certain polynomial maps.

Provides explicit relations between bifurcation sets 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.