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Volume 55, Issue 4, 2019, pp. 811–834
DOI: 10.4171/PRIMS/55-4-6

Published online: 2019-10-21

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

Tat Thang Nguyen[1], Phú Phát Phạm[2] and Tiến-Sơn Phạm[3]

(1) Vietnam Academy of Science and Technology, Hanoi, Vietnam
(2) University of Dalat, Vietnam
(3) University of Dalat, Vietnam

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.

Keywords: Singularities of polynomial functions, bifurcation set, global monodromies, brations, tangency values, Newton polyhedra

Nguyen Tat Thang, Phạm Phú Phát, Phạm Tiến-Sơn: Bifurcation Sets and Global Monodromies of Newton Nondegenerate Polynomials on Algebraic Sets. Publ. Res. Inst. Math. Sci. 55 (2019), 811-834. doi: 10.4171/PRIMS/55-4-6