Bifurcation Sets and Global Monodromies of Newton Nondegenerate Polynomials on Algebraic Sets

Abstract

Let $S\subset \mathbb{C}^n$ be a nonsingular algebraic set and $f \colon \mathbb{C}^n \to \mathbb{C}$ be a polynomial function. It is well known that the restriction $f|\_S \colon S \to \mathbb{C}$ of $f$ on $S$ is a locally trivial fibration outside a finite set $B(f|S) \subset \mathbb{C}.$ In this paper we give an explicit description of a finite set $T\infty(f|\_S) \subset \mathbb{C}$ such that $B(f|\_S) \subset K\_0(f|S) \cup T\infty(f|\_S),$ where $K\_0(f|\_S)$ denotes the set of critical values of the $f|S.$ Furthermore, $T\infty(f|\_S)$ is contained in the set of critical values of certain polynomial functions provided that the $f|S$ is Newton nondegenerate at infinity. Using these facts, we show that if ${f\_t}{t \in \[0, 1]}$ is a family of polynomials such that the Newton polyhedron at infinity of $f\_t$ is independent of $t$ and the $f\_t|\_S$ is Newton nondegenerate at infinity, then the global monodromies of the $f\_t|\_S$ are all isomorphic.