Zeta functions for germs of meromorphic functions, and Newton diagrams

Abstract

For a germ of a meromorphic function f = P Q , we offer notions of the monodromy operators at zero and at infinity. If the holomorphic functions P and Q are non-degenerated with respect to their Newton diagrams, we give an analogue of the formula of Varchenko for the zeta-functions of these monodromy operators.

Deformations of polynomials and their zeta-functions

“…An interesting case is a linear family of polynomials and respectively linear families of holomorphic germs. In somewhat other terms this case was treated in the study of meromorphic germs elaborated for study of polynomial maps: [4,5]. Let s σ = f + σg be a linear family of holomorphic germs (the family of the zero level sets of s σ is a pencil).…”

Milnor fibrations of meromorphic functions

“…According to [4,Lemma 1], the diffeomorphism class of the noncompact n-complex manifold M c F does not depend on ε, and the isomorphism class of the fibration φ c does not depend on ε and δ. As shown in [14], this is in fact an immediate consequence of Lê's fibration theorem in [6] applied to the pencil {f − tg = 0}.…”

“…It is thus natural to ask whether one has for meromorphic mapgerms fibrations of Milnor type (1), and if so, how these are related to those of the Milnor-Lê type (2) studied (for instance) in [4,5,14]. The first of these questions was addressed in [12,1,13] from two different viewpoints, while the answer to the second question is the bulk of this article.…”

“…See for instance [4,5], or Tibar's book [14] and the references in it. Of course the "Milnor tubes" B ε ∩ F −1 (∂D δ ) in this case are not actual tubes in general, since they may contain 0 ∈ U in their closure.…”

Zeta functions and oscillatory integrals for meromorphic functions

“…S. Gusein-Zade, I. Luengo and A. Melle-Hernández have studied the complex monodromy (and A'Campo zeta functions attached to it) of meromorphic functions, see e.g. [25], [26], [27]. This work drives naturally to ask about the existence of local zeta functions with poles related with the monodromies studied by the mentioned authors.…”