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

### Tat Thang Nguyen

Vietnam Academy of Science and Technology, Hanoi, Vietnam### Phú Phát Phạm

University of Dalat, Vietnam### Tiến-Sơn Phạm

University of Dalat, Vietnam

## 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.

## Cite this article

Tat Thang Nguyen, Phú Phát Phạm, Tiến-Sơn Phạm, Bifurcation Sets and Global Monodromies of Newton Nondegenerate Polynomials on Algebraic Sets. Publ. Res. Inst. Math. Sci. 55 (2019), no. 4, pp. 811–834

DOI 10.4171/PRIMS/55-4-6