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For an affine toric variety X we compute the Poincare series of the multi-index filtration defined by a finite number of monomial divisorial valuations on the ring O_{X,0}. We give an alternative description of the Poincare series as an…

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…

To study singularities on complex varieties we study Poincar\'e series of filtrations that are defined by discrete valuations on the local ring at the singularity. In all previous papers on this topic one poses restrictions on the centers…

The geometric motivic Poincar\'e series of a germ $(S,0)$ of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through $(S,0)$. Denef and Loeser proved that this series has a rational…

We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible…

In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincar\'e series was…

Earlier, there was computed the Poincar\'e series of a valuation or of a collection of valuations on the ring of germs of holomorphic functions in two variables. For a collection of several plane curve valuations it appeared to coincide…

For a subfield $\K$ of the field $\C$ of complex numbers, we consider curve and divisorial valuations on the algebra $\K[[x,y]]$ of formal power series in two variables with the coeficients in $\K$. We compute the semigroup Poincar\'e…

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…

Earlier, there were defined two generalized (``motivic'') versions of the Poincar\'e 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…

A relation is proved between the Poincar\'{e} series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity…

In the author's paper ''Poincar\'{e} series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity'' a relation is proved between the Poincar\'{e} series of the coordinate algebra of a two-dimensional quasihomogeneous…

In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincar\'e series of the invariants and concomitants of the general linear Lie supergroup…

We define a multi-index filtration on the ring of germs of functions on a hypersurface singularity associated with its Newton diagram and compute the multivariable Poincar\'e series of this filtration in some cases.

Earlier the authors considered and, in some cases, computed Poincare series of two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular on the complex plane). A…

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 Poincar\'e series, the generalized Poincar\'e series, and the…

We study the counting function of topological Poincar\'e series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with…

We consider divisorial filtration on the rings of functions on hypersurface singularities associated with Newton diagrams and their analogues for plane curve singularities. We compute the multi-variable Poincar\'e series for the latter…

The Poincare series of a multi-index filtration on the ring of germs of functions can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring. Here this integral is considered with…

Earlier it was described to which extent the Alexander polynomial in several variables of an algebraic link in the Poincar\'e sphere determines the topology of the link. It was shown that, except some explicitly described cases, the…