Poincare series of collections of plane valuations

In earlier papers there were given formulae for the Poincare series of multi-index filtrations on the ring of germs of functions of two variables defined by collections of valuations corresponding to (reducible) plane curve singularities and by collections of divisorial ones. It was shown that the Poincare series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincare series of a general collection of valuations on the ring of germs of functions of two variables centred at the origin and prove a generalization of the statement that the Poincare series determines the topology of the collection

Integration with respect to Euler characteristic over the projectivization of the space of functions and the Alexander polynomial of a plane curve singularity

We discuss a notion of integration with respect to the Euler characteristic in the projectivization \P{\cal O}_{\C^n,0} of the ring {\cal O}_{\C^n,0} of germs of functions on $C^n$ and show that the Alexander polynomial and the zeta-function of a plane curve singularity can be expressed as certain integrals over \P{\cal O}_{\C^2,0} with respect to the Euler characteristic

On Poincare series of filtrations on equivariant functions of two variables

Let a finite group $G$ act on the complex plane $({\Bbb C}^2, 0)$. We consider multi-index filtrations on the spaces of germs of holomorphic functions of two variables equivariant with respect to 1-dimensional representations of the group $G$ defined by components of a modification of the complex plane ${\Bbb C}^2$ at the origin or by branches of a $G$-invariant plane curve singularity $(C,0)\subset({\Bbb C}^2,0)$. We give formulae for the Poincare series of these filtrations. In particular, this gives a new method to obtain the Poincare series of analogous filtrations on the rings of germs of functions on quotient surface singularities

On the topological type of a set of plane valuations with symmetries

Let $\{C_i : i=1,\ldots,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $\Delta^{L}(\{t_{a i}\})$ be the Alexander polynomial in $r\vert G\vert$ variables of the algebraic link $(\bigcup\limits_{i=1}^{r}\bigcup\limits_{a\in G}a C_i )\cap S^3_{\varepsilon}$ and let $\zeta(t_1,\ldots, t_r) = \Delta^{L}(t_1,\ldots,t_1,t_2,\ldots,t_2, \ldots,t_r,\ldots,t_r)$ with $\vert G\vert$ identical variables in each group. (If $r=1$, $\zeta(t)$ is the monodromy zeta function of the function germ $\prod\limits_{a\in G} a^*f$, where $f=0$ is an equation defining the curve $C_1$.) We prove that $\zeta(t_1,\ldots, t_r)$ determines the topological type of the link $L$. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincar\'e series of a set of valuations

Equivariant Poincar\'e series and topology of valuations

The equivariant with respect to a finite group action Poincar\'e series of a collection of $r$ valuations was defined earlier as a power series in $r$ variables with the coefficients from a modification of the Burnside ring of the group. Here we show that (modulo simple exceptions) the equivariant Poincar\'e series determines the equivariant topology of the collection of valuations.Comment: arXiv admin note: text overlap with arXiv:1405.300

On Poincar\'e series of filtrations

In this survey one discusses the notion of the Poincar\'e series of multi-index filtrations, an alternative approach to the definition, a method of computation of the Poincar\'e series based on the notion of integration with respect to the Euler characteristic (or rather on an infinite-dimensional version of it), generalizations of the notion of the multi-variable Poincar\'e series based on the notion of the motivic integration, and relations of the latter ones with some zeta functions over finite fields and with generating series of Heegaard-Floer homologies of algebraic links

Universal abelian covers of rational surface singularities and multi-index filtrations

In previous papers, there were computed the Poincare series of some (multi-index) filtrations on the ring of germs of functions on a rational surface singularity. These Poincare series were written as the integer parts of certain fractional power series, an interpretation of whom was not given. Here we show that, up to a simple change of variables, these fractional power series are specializations of the equivariant Poincare series for filtrations on the ring of germs of functions on the universal abelian cover of the surface singularity. We compute these equivariant Poincare series

An equivariant Poincar\'e series of filtrations and monodromy zeta functions

We define a new equivariant (with respect to a finite group $G$ action) version of the Poincar\'e series of a multi-index filtration as an element of the power series ring ${\widetilde{A}}(G)[[t_1, \ldots, t_r]]$ for a certain modification ${\widetilde{A}}(G)$ of the Burnside ring of the group $G$. We give a formula for this Poincar\'e series of a collection of plane valuations in terms of a $G$-resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve singularity defined by a $G$-invariant function germ, in the majority of cases this equivariant Poincar\'e series determines the corresponding equivariant monodromy zeta functions defined earlier

The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors

To a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincare series of this filtration turnes out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincare series of the corresponding multi-index filtration similar to the one associated to plane germs. Here we show that the Poincare series of a set of divisorial valuations on the ring of germs of functions of two variables defines "the topology of the set of the divisors" in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which proves that the Alexander polynomial is equivalent to the embedded topology

Integrals with respect to the Euler characteristic over spaces of functions and the Alexander polynomial

We discuss some formulae which express the Alexander polynomial (and thus the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on the curve. One of them describes the coefficients of the Alexander polynomial or of zeta-function of the monodromy transformation as Euler characteristics of some explicitly constructed spaces. For the Alexander polynomial these spaces are complements to arrangements of projective hyperplanes in projective spaces. For the zeta-function they are disjoint unions of such spaces. Under the influence of a result by J.Denef and F.Loeser it was understood that this results are connected with the notion of the motivic integration or rather with its version (in some sense a dual one) where the space of arcs is substituted by the space of functions. The aim of this paper is to discuss the notion of the integral with respect to the Euler characteristics (or with respect to the generalized Euler characteristic) over the space of functions (or over its projectivization) and its connection with the formulae for the coefficients of the Alexander polynomial and of the zeta-function of the monodromy transformation as Euler characteristics of some spaces. The paper will be published in Proceedings of the Steklov Mathematical Institute