A. Melle-Hernández

Zeta-functions for germs of meromorphic functions and Newton diagrams

Let f be a meromorphic function germ on (Cn+1, 0); that is, f = P/Q, where P,Q: (Cn+1, 0)! (C, 0) are holomorphic germs. The authors introduce a notion of Milnor fibers and monodromy operators of the germ f around zero and infinity. Based on their previous work [Comment. Math. Helv. 72 (1997), no. 2, 244–256; MR1470090 (98j:32043)] they write down formulas for the zetafunctions of the monodromy operators in terms of partial resolutions of a singularity. In the case where P and Q are non-degenerate relative to their Newton’s diagrams an analog of the formula from [A. N. Varchenko, Invent. Math. 37 (1976), no. 3, 253–262; MR0424806 (54 #12764)] for zeta functions of monodromy operators is obtained. In conclusion, two interesting examples with f = (x3 −xy)/y and f = (xyz +xp +yq +zr)/(xd +yd +zd) are discussed in detail.

On the Zeta-function of a polynomial at infinity

A polynomial function defines a locally trivial fibre bundle over the complement to a finite set in the target C. Objects connected with this fibration (say, monodromy operators and, in particular, the monodromy operator of the polynomial at infinity) are in some sense global. The idea of the paper is to localize computations of the zeta-functions of monodromy transformations for a polynomial, i.e., to express them in local terms. It is done with the use of the notion of Milnor fibres of the germ of a meromorphic function and the methods of calculation of the corresponding zeta-functions elaborated by the authors. 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.

Integration over spaces of nonparametrized arcs and motivic versions of the monodromy zeta function

Notions of integration of motivic type over the space of arcs factorized by the natural C*-action and over the space of nonparametrized arcs (branches) are developed. As an application, two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on ℂd are given that correspond to these notions. Another key ingredient in the construction of these motivic versions of the zeta function is the use of the so-called power structure over the Grothendieck ring of varieties introduced by the authors.

On Some Conjectures About Free and Nearly Free Divisors

In this paper we provide infinite families of non-rational irreducible free divisors or nearly free divisors in the complex projective plane. Moreover, their corresponding local singularities can have an arbitrary number of branches. All these examples contradict some of the conjectures proposed by Dimca and Sticlaru. Our examples say nothing about the most remarkable conjecture by A. Dimca and G. Sticlaru, which predicts that every rational cuspidal plane curve is either free or nearly free.

Yano’s conjecture for two-Puiseux-pair irreducible plane curve singularities

In 1982, Yano proposed a conjecture predicting the

Generating series of classes of Hilbert schemes of points on orbifolds

b

Power structure over the Grothendieck ring of maps

-exponents of an irreducible plane curve singularity which is generic in its equisingularity class. In this article we prove the conjecture for the case of two Puiseux pairs and monodromy with distinct eigenvalues. The hypothesis on the monodromy implies that the

High-school algebra of the theory of dicritical divisors: Atypical fibers for special pencils and polynomials

b

Partial resolutions and the zeta-function of a singularity

-exponents coincide with the opposite of the roots of the Bernstein polynomial, and we compute the roots of the Bernstein polynomial.

Milnor number at infinity, topology and Newton boundary of a polynomial function

The notion of power structure over the Grothendieck ring of complex quasi-projective varieties is used for describing generating series of classes of Hilbert schemes of zero-dimensional subschemes (“fat points”) on complex orbifolds.