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

TL;DR

This paper explicitly describes the bifurcation set of polynomial functions on algebraic sets, especially under Newton non-degeneracy at infinity, and shows the invariance of global monodromies in certain polynomial families.

Contribution

It provides an explicit description of the bifurcation set for polynomial functions on algebraic sets and establishes monodromy invariance under Newton non-degeneracy conditions.

Findings

Explicit description of bifurcation sets including critical and asymptotic values.

Identification of conditions ensuring monodromy invariance in polynomial families.

Demonstration of invariance of global monodromies for Newton non-degenerate polynomials.

Abstract

Let $S\subset \mathbb{C}^n$ be a non-singular 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 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.