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POINCARÉ SERIES FOR SEVERAL PLANE DIVISORIAL VALUATIONS

Published online by Cambridge University Press:  04 July 2003

F. Delgado  and
S. M. Gusein-Zade
F. Delgado
Affiliation:
Department of Algebra, Geometry and Topology, University of Valladolid, 47005 Valladolid, Spain (fdelgado@agt.uva.es)
S. M. Gusein-Zade
Affiliation:
Faculty of Mathematics and Mechanics, Moscow State University, Moscow 119899, Russia (sabir@mccme.ru)
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Abstract

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We compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.

AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

Keywords

multi-index filtrationsPoincaréseriesdivisorial valuationsgerms of functions of two variables
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 2 , June 2003 , pp. 501 - 509
Copyright
Copyright © Edinburgh Mathematical Society 2003