Hilbert function, generalized Poincaré series and topology of plane valuations
type: Item
instance of: scholarly article
title: Hilbert function, generalized Poincaré series and topology of plane valuations
zbMATH Open document ID: 1325.14010
published in: Monatshefte für Mathematik
publication date: 2014-08-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1303.3648
review text: Hilbert function, Hilbert series, Poincaré series and generalized Poincaré series attached to multi-index filtrations are objects which have been rather studied in a large series of interesting articles by the authors of this paper. This one keeps the same line and the authors prove that for filtrations given by what they call order functions (objects close to valuations that need not to satisfy \(\nu(fg)=\nu(f) + \nu(g)\)), Hilbert function, Hilbert series and generalized Poincaré series are equivalent in the sense that one of them determines the others. Notice that this is not the case of Poincaré series. The paper is also devoted to prove that the topological type of finite families of plane valuations (excluding curve valuations of rank 2) is determined by the mentioned generalized Poincaré series (and therefore by the Hilbert function).
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 16W70
zbMATH DE Number: 6330893
zbMATH Keywords: filtrations
zbMATH Keywords: Hilbert functions
zbMATH Keywords: Poincaré series
zbMATH Keywords: plane valuations
author: Antonio Campillo
author: F. Delgado
author: Sabir M. Gusein-Zade
reviewed by: Carlos Galindo
MaRDI profile type: Publication
OpenAlex ID: W2005287138
arXiv ID: 1303.3648
cites work: THE ALEXANDER POLYNOMIAL OF A PLANE CURVE SINGULARITY AND INTEGRALS WITH RESPECT TO THE EULER CHARACTERISTIC
cites work: The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors
cites work: Multi-index filtrations and generalized Poincaré series
cites work: POINCARÉ SERIES OF COLLECTIONS OF PLANE VALUATIONS
cites work: Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings
cites work: Poincaré series of resolutions of surface singularities
cites work: The semigroup of values of a curve singularity with several branches
cites work: Germs of arcs on singular algebraic varieties and motivic integration
cites work: Lattice and Heegaard Floer Homologies of Algebraic Links
cites work: Motivic zeta functions for curve singularities
cites work: Valuations in Function Fields of Surfaces
cites work: Classification of isolated algebraic singularities by their Alexander polynomials
DOI: 10.1007/S00605-013-0547-5
Recommended article: Equivariant Poincaré series and topology of valuations
Recommended article: On Hilbert functions and cohomology
Recommended article: Q3592297
Recommended article: Hilbert series for equivariant mappings restricted to invariant hyperplanes
Recommended article: Q4295735
Recommended article: Q3735861
Recommended article: Q3027137
Recommended article: POINCARÉ SERIES OF COLLECTIONS OF PLANE VALUATIONS
Recommended article: Graded Multiplicity Theory and Hilbert Functions
Recommended article: Hilbert-Kunz functions of a Hirzebruch surface
description: scientific article; zbMATH DE number 6330893