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

2018/03/26 by Nguyen, Tat Thang, Pham, Phu Phat, Pham, Tien-Son
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.09654

Abstract

Let S⊂ ℂⁿ be a non-singular algebraic set and f \colon ℂⁿ → ℂ be a polynomial function. It is well-known that the restriction f|_S \colon 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) ⊂ K₀(f|_S) ∪ T_∞(f|_S), where K₀(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 non-degenerate 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 non-degenerate at infinity, then the global monodromies of the f_t|_S are all isomorphic.

Related

• On Newton non-degeneracy of polynomial mappings
• Newton polygons and families of polynomials
• The tropical non-properness set of a polynomial map
• The bifurcation set of a real polynomial function of two variables and Newton polygons of singularities at infinity
• Newton numbers, vanishing polytopes and algebraic degrees