Zeta functions for germs of meromorphic functions, and Newton diagrams (original) (raw)

On atypical values and local monodromies of meromorphic functions

Arxiv preprint math/ …, 1998

A meromorphic function on a compact complex analytic manifold defines a C ∞ locally trivial fibration over the complement of a finite set in the projective line CP 1. We describe zeta-functions of local monodromies of this fibration around atypical values. Some applications to polynomial functions on C n are described. §1.-Introduction We want to consider fibrations defined by meromorphic functions. In order to have more general statements we prefer to use the notion of a meromorphic function slightly different from the standard one. Let M be an n-dimensional compact complex analytic manifold.

Monodromy eigenvalues and zeta functions with differential forms

Advances in Mathematics, 2007

For a complex polynomial or analytic function f , there is a strong correspondence between poles of the so-called local zeta functions or complex powers |f | 2s ω, where the ω are C ∞ differential forms with compact support, and eigenvalues of the local monodromy of f . In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(2π √ −1s 0 ), where s 0 is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.

The ubiquitous ζ-function and some of its ‘usual’ and ‘unusual’ meromorphic properties

Journal of Physics A: Mathematical and Theoretical, 2008

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of ζ-functions associated to conic manifolds proved in [37]. In particular, we show that the meromorphic extensions of these ζfunctions have, in general, countably many logarithmic branch cuts on the nonpositive real axis and unusual locations of poles with arbitrarily large multiplicity. Moreover, we give a precise algebraic-combinatorial formula to compute the coefficients of the leading order terms of the singularities.

Monodromy eigenvalues are induced by poles of zeta functions: the irreducible curve case

Bulletin of the London Mathematical Society, 2010

The 'monodromy conjecture' for a hypersurface singularity f predicts that a pole of its topological (or related) zeta function induces one of its monodromy eigenvalues. However, in general only a few eigenvalues are obtained this way. The second author proposed to consider zeta functions associated with the hypersurface and with a differential form and raised the following question. Can one find a list of differential forms ω i such that any pole of the zeta function of f and an ω i induces a monodromy eigenvalue of f , and such that all monodromy eigenvalues of f are obtained this way? Here we provide an affirmative answer for an arbitrary irreducible curve singularity f .

On the topology of germs of meromorphic functions and applications

Eprint Arxiv Math 9905127, 1999

Germs of meromorphic functions has recently become an object of study in singularity theory. T. Suwa ([11]) described versal deformations of meromorphic germs. V.I. Arnold ([1]) classified meromorphic germs with respect to certain equivalence relations. The authors ([4]) started a study of topological properties of meromorphic germs. Some applications of the technique developed in [4] were described in [5] and [6]. In [4] the authors elaborated notions and technique which could be applied to compute such invariants of polynomials as Euler characteristics of fibres and zetafunctions of monodromy transformations associated with a polynomial (see [5]). Some crucial basic properties of the notions related to the topology of meromorphic germs were not discussed there. This has produced some lack of understanding of the general constructions. The aim of this note is to partially fill in this gap. At the same time we describe connections with some previous results and generalizations of them. A polynomial P in n + 1 complex variables defines a map P from the affine complex space C n+1 to the complex line C. It is well known that the map P is a C ∞-locally trivial fibration over the complement to a finite set in the line C. The smallest of such sets is called the bifurcation set or the set of atypical values of the polynomial P. One is interested in describing the topology of the fibre of this fibration and its behaviour under monodromy transformations corresponding to loops around atypical values of the polynomial P. The monodromy transformation corresponding to a circle of big radius which contains all atypical values (the monodromy transformation of the polynomial P at infinity) is of particular interest. The initial idea was to reduce calculation of the zeta-function of the monodromy transformation at infinity (and thus of the Euler characteristic of the generic fibre) of the polynomial P to local problems associated to different points at infinity, i.e., at the infinite hyperplane CP n ∞ in the projective compactification CP n+1 of the affine space C n+1. The possibility of such a localization for holomorphic germs was used in [3]. This localization can be expressed in terms of an integral with respect to the Euler characteristic, a notion introduced by the school of V.A. Rokhlin ([12]). However the results are not apply directly to a polynomial function since at a point of the infinite hyperplane CP n ∞ a polynomial function defines not a Key words and phrases. Germs of meromorphic functions, Milnor fibre, atypical values.

On Monodromy of Generalized Analytic Functions

Journal of Mathematical Sciences, 2006

We discuss a number of topics concerned with certain boundary-value problems in the context of generalized analytic functions. Solution of the classical Riemann-Hilbert problem and the linear conjugation problem for analytic functions is described in appropriate function classes and the same scheme is applied to generalized analytic functions and vectors. In particular, we describe solution of the Riemann-Hilbert problem for generalized analytic functions and obtain an explicit analytic presentation of monodromy matrices in the case of generalized analytic vectors.

On the zeta-function of a polynomial at infinity

arXiv (Cornell University), 1998

We use the notion of Milnor fibres of the germ of a meromorphic function and the method of partial resolutions for a study of topology of a polynomial map at infinity (mainly for calculation of the zeta-function of a monodromy). It gives effective methods of computation of the zeta-function for a number of cases and a criterium for a value to be atypical at infinity. §1.-Introduction The main idea of the paper is to bring together methods of [7] and [8] for computing the zeta-function of the monodromy at infinity of a polynomial. Let P be a complex polynomial in (n + 1) variables. It defines a map from C n+1 to C which also will be denoted by P. It is known ([13]) that there exists a finite set B(P) ⊂ C such that the map P is a C ∞ locally trivial fibration over its complement. The monodromy transformation h of this fibration corresponding to the loop z 0 • exp(2πiτ) (0 ≤ τ ≤ 1) with z 0 big enough is called the geometric monodromy at infinity of the polynomial P. Let h * be its action in the homology groups of the fibre (the level set) {P = z 0 }. Definition. The zeta-function of the monodromy at infinity of the polynomial P is the rational function ζ P (t) = q≥0 {det [ id − t h * | H q ({P =z 0 };C) ]} (−1) q. Remark 1. We use the definition from [2], which means that the zeta-function defined this way is the inverse of that used in [1]. The degree of the zeta-function (the degree of the numerator minus the degree of the denominator) is equal to the Euler characteristic χ P of the (generic) fibre {P = z 0 }. Formulae for the zeta-functions at infinity for certain polynomials were given in particular in [6], [9].

Deformations of polynomials and their zeta-functions

Journal of Mathematical Sciences, 2007

For an analytic in σ ∈ (C, 0) family P σ of polynomials in n variables there is defined a monodromy transformation h of the zero level set V σ = {P σ = 0} for σ = 0 small enough. The zeta function of this monodromy transformation is written as an integral with respect to the Euler characteristic of the corresponding local data. This leads to a study of deformations of holomorphic germs and their zeta functions. We show some examples of computations with the use of this technique. *

The Lerch zeta function II. Analytic continuation

Forum Mathematicum, 2000

This is the second of a series of four papers that study algebraic and analytic structures associated with the Lerch zeta function. The Lerch zeta function ζ(s, a, c) := ∞ n=0 e 2πina (n+c) s was introduced by Lipschitz in 1857, and is named after Lerch, who showed in 1887 that it satisfied a functional equation. Here we analytically continue ζ(s, a, c) as a function of three complex variables. We show that it is welldefined as a multivalued function on the manifold M := {(s, a, c) ∈ C × (C Z) × (C Z)}, and that this analytic continuation becomes single-valued on the maximal abelian cover of M. We compute the monodromy functions describing the multivalued nature of this function on M, and determine various of its properties. Contents 1. Introduction 1 2. Summary of results 4 3. Functional equations 8 4. Analytic continuation and monodromy functions 10 5. Differential-difference operators and Lerch monodromy functions 20 6. Functional equations and monodromy functions 23 7. Lerch monodromy vector spaces 24 8. Extended analytic continuation 26 9. Concluding remarks 27 References 28

An equivariant version of the monodromy zeta function

American Mathematical Society Translations: Series 2, 2008

We offer an equivariant version of the classical monodromy zeta function of a singularity as a series with coefficients from the Grothendieck ring of finite G-sets tensored by the field of rational numbers. Main two ingredients of the definition are equivariant Lefschetz numbers and the λ-structure on the Grothendieck ring of finite G-sets. We give an A'Campo type formula for the equivariant zeta function.