Bifurcation sets and global monodromies of Newton non-degenerate polynomials on algebraic sets

Abstract

Let $S\subset {\mathbb{C}}^n$ be a non-singular algebraic set and $f:{\mathbb{C}}^n\to \mathbb{C}$ be a polynomial function. It is well-known that the restriction $f{|}_S:S\to \mathbb{C}$ 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 non-degenerate 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 non-degenerate at infinity, then the global monodromies of the $f_t{|}_S$ are all isomorphic.