JournalsprimsVol. 55, No. 4pp. 811–834

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

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Abstract

Let SCnS\subset \mathbb{C}^n be a nonsingular algebraic set and f ⁣:CnCf \colon \mathbb{C}^n \to \mathbb{C} be a polynomial function. It is well known that the restriction fS ⁣:SCf|_S \colon S \to \mathbb{C} of ff on SS is a locally trivial fibration outside a finite set B(fS)C.B(f|_S) \subset \mathbb{C}. In this paper we give an explicit description of a finite set T(fS)CT_\infty(f|_S) \subset \mathbb{C} such that B(fS)K0(fS)T(fS),B(f|_S) \subset K_0(f|_S) \cup T_\infty(f|_S), where K0(fS)K_0(f|_S) denotes the set of critical values of the fS.f|_S. Furthermore, T(fS)T_\infty(f|_S) is contained in the set of critical values of certain polynomial functions provided that the fSf|_S is Newton nondegenerate at infinity. Using these facts, we show that if {ft}t[0,1]\{f_t\}_{t \in [0, 1]} is a family of polynomials such that the Newton polyhedron at infinity of ftf_t is independent of tt and the ftSf_t|_S is Newton nondegenerate at infinity, then the global monodromies of the ftSf_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