arXiv subjects

S. M. Gusein-Zade

Publications and source records attributed to S. M. Gusein-Zade.

At least 19 recordsLinked to original sources

arXiv2026

On a generalized Poincaré series of plane valuations

Earlier, there were defined two generalized (``motivic'') versions of the Poincaré series of a collection of plane valuations on the algebra ${\mathcal O}_{{\mathbb C}^2,0}$ of germs of holomorphic functions in two variables. One of them was defined as an integral with respect to the generalized Euler characteristic over the projectivization of the extended semigroup of the collection. One has a natural version of it for valuations on the algebra ${\mathcal E}_{{\mathbb K}^2,0}$ of germs of holomorphic functions in two variables whose Taylor coefficients are from a fixed subfield ${\mathbb K}$ of the field ${\mathbb C}$ of complex numbers. In this setting the usual Poincaré series were computed for one plane curve or divisorial valuation on ${\mathcal E}_{{\mathbb K}^2,0}$. We give equations for the corresponding generalized Poincaré series.

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arXiv2023

On real analogues of the Poincaré series

There exist several equivalent equations for the Poincaré series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poincaé series of a collection of valuations in the real setting (i.e., on the ring of germs of functions on a real analytic variety), compute them for the case of one curve valuation on the plane and discuss some of their properties.

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arXiv2021

Weil-Poincaré series and topology of collections of valuations on rational double points

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It was shown that, except some explicitly described cases, the Alexander polynomial of an algebraic link determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the corresponding set of curve valuations. The latter one can be defined as an integral over the space of divisors on the $E_8$-singularity. Here we consider a similar integral for rational double point surface singularities over the space of Weil divisors called the Weil-Poincaré series. We show that, except a few explicitly described cases the Weil-Poincaré series of a collection of curve valuations on a rational double point surface singularity determines the topology of the corresponding link. We give analogous statements for collections of divisorial valuations.

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arXiv2019

On simple $Z_2$-invariant and corner function germs

V.I.Arnold has classified simple (i.e. having no modules for the classification) singularities (function germs), and also simple boundary singularities (function germs invariant with respect to the action $σ(x_1; y_1, \ldots, y_n)=(-x_1; y_1, \ldots, y_n)$ of the group $Z_2$. In particular, it was shown that a function germ (respectively a boundary singularity germ) is simple if and only if the intersection form (respectively the restriction of the intersection form to the subspace to anti-invariant cycles) of a germ in $3+4s$ variables stable equivalent to the one under consideration is negative definite and if and only if the (equivariant) monodromy group on the corresponding subspace is finite. We formulate and prove analogues of these statements for function germs invariant with respect to an arbitrary action of the group $Z_2$ and also for corner singularities.

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arXiv2019

Generalized orbifold Euler characteristics on the Grothendieck ring of varieties with actions of finite groups

The notion of the orbifold Euler characteristic came from physics at the end of 80's. There were defined higher order versions of the orbifold Euler characteristic and generalized ("motivic") versions of them. In a previous paper the authors defined a notion of the Grothendieck ring $K_0^{\rm fGr}(Var)$ of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from $K_0^{\rm fGr}(Var)$ to the Grothendieck ring $K_0(Var)$ of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

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arXiv2019

Index of a singular point of a vector field or of a 1-form on an orbifold

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincaré--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier there was defined a notion of the universal Euler characteristic of an orbifold. It takes values in a ring R, as an abelian group freely generated by the generators, corresponding to the isomorphism classes of finite groups. Here we define the universal index of an isolated singular point of a vector field or of a 1-form on an orbifold as an element of the ring R. For this index, an analogue of the Poincaré-Hopf theorem holds.

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arXiv2018

The universal Euler characteristic of V-manifolds

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with some finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. Here we discuss a universal additive topological invariant of V-manifolds: the universal Euler characterictic. It takes values in the ring generated (as a Z-module) by isomorphism classes of finite groups. We also consider the universal Euler characteristic on the class of locally closed equivariant unions of cells in equivariant CW-complexes. We show that it is a universal additive invariant satisfying a certain "induction relation". We give Macdonald type equations for the universal Euler characteristic for V-manifolds and for cell complexes of the described type.

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arXiv2018

Algebraic links in the Poincaré sphere and the Alexander polynomials

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the surface $S=\{(z_1,z_2,z_3)\in {\mathbb C}^3: z_1^5+z_2^3+z_3^2=0\}$ with the 5-dimensional sphere ${\mathbb S}_{\varepsilon}^5=\{(z_1,z_2,z_3)\in {\mathbb C}^3: \vert z_1\vert^2+\vert z_2\vert^2+\vert z_3\vert^2=\varepsilon^2\}$. An algebraic link in the Poincaré sphere is the intersection of a germ $(C,0)\subset (S,0)$ of a complex analytic curve in $(S,0)$ with the sphere ${\mathbb S}_{\varepsilon}^3$ of radius $\varepsilon$ small enough. Here we discuss to which extend the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. We show that, if the strict transform of a curve on $(S,0)$ does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding $E_8$-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. Alexander polynomial of an algebraic link in the Poincaré sphere coincides with the Poincaré series of the filtration defined by the corresponding curve valuations. We show that, under conditions similar for those for curves, the Poincaré series of a collection of divisorial valuations determines the combinatorial type of the minimal resolution of the collection.

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arXiv2017

Grothendieck ring of varieties with finite groups actions

We define a Grothendieck ring of varieties with finite groups actions and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural $λ$-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized ("motivic") Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher order Euler characteristics of wreath products.

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arXiv2016

On equivariant versions of higher order orbifold Euler characteristics

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher order versions. Here we suggest a way to merge the two approaches together defining (in a certain setting) higher order Euler characteristics with values in the Burnside ring of a group. We give Macdonald type equations for these invariants. We also offer generalized ("motivic") versions of these invariants and formulate Macdonald type equations for them as well.

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arXiv2016

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 $Δ^{L}(\{t_{a i}\})$ 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 $ζ(t_1,\ldots, t_r) = Δ^{L}(t_1,\ldots,t_1,t_2,\ldots,t_2, \ldots,t_r,\ldots,t_r)$ with $\vert G\vert$ identical variables in each group. (If $r=1$, $ζ(t)$ 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 $ζ(t_1,\ldots, t_r)$ determines the topological type of the link $L$. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.

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arXiv2015

Equivariant Poincaré series and topology of valuations

The equivariant with respect to a finite group action Poincaré 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 Poincaré series determines the equivariant topology of the collection of valuations.

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arXiv2015

On Poincaré series of filtrations

In this survey one discusses the notion of the Poincaré series of multi-index filtrations, an alternative approach to the definition, a method of computation of the Poincaré 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 Poincaré 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.

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arXiv2014

An equivariant Poincaré series of filtrations and monodromy zeta functions

We define a new equivariant (with respect to a finite group $G$ action) version of the Poincaré 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 Poincaré 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 Poincaré series determines the corresponding equivariant monodromy zeta functions defined earlier.

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arXiv2014

Higher order orbifold Euler characteristics for compact Lie group actions

We generalize the notions of the orbifold Euler characteristic and of the higher order orbifold Euler characteristics to spaces with actions of a compact Lie group. This is made using the integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the k-th order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by H. Tamanoi for finite group actions and by C. Farsi and Ch. Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.

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arXiv2013

Higher order generalized Euler characteristics and generating series

For a complex quasi-projective manifold with a finite group action, we define higher order generalized Euler characteristics with values in the Grothendieck ring of complex quasi-projective varieties extended by the rational powers of the class of the affine line. We compute the generating series of generalized Euler characteristics of a fixed order of the Cartesian products of the manifold with the wreath product actions on them.

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arXiv2013

Hilbert function, generalized Poincaré series and topology of plane valuations

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.

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arXiv2013

On an equivariant version of the zeta function of a transformation

Earlier the authors offered an equivariant version of the classical monodromy zeta function of a G-invariant function germ with a finite group G as a power series with the coefficients from the Burnside ring of the group G tensored by the field of rational numbers. One of the main ingredients of the definition was the definition of the equivariant Lefschetz number of a G-equivariant transformation given by W.Lück and J.Rosenberg. Here we offer another approach to a definition of the equivariant Lefschetz number of a transformation and describe the corresponding notions of the equivariant zeta function. This zeta-function is a power series with the coefficients from the Burnside ring of the group G. We give an A'Campo type formula for the equivariant monodromy zeta function of a function germ in terms of a resolution. Finally we discuss orbifold versions of the Lefschetz number and of the monodromy zeta function corresponding to the two equivariant ones.

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