# 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 $S⊂C_{n}$ be a nonsingular algebraic set and $f:C_{n}→C$ be a polynomial function. It is well known that the restriction $f∣_{S}:S→C$ of $f$ on $S$ is a locally trivial fibration outside a finite set $B(f∣_{S})⊂C.$ In this paper we give an explicit description of a finite set $T_{∞}(f∣_{S})⊂C$ such that $B(f∣_{S})⊂K_{0}(f∣_{S})∪T_{∞}(f∣_{S}),$ where $K_{0}(f∣_{S})$ denotes the set of critical values of the $f∣_{S}.$ Furthermore, $T_{∞}(f∣_{S})$ 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 ${f_{t}}_{t∈[0,1]}$ 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