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POINCARÉ SERIES OF COLLECTIONS OF PLANE VALUATIONS

A. CAMPILLO

University of Valladolid, Department of Algebra, Geometry and Topology, 47011 Valladolid, Spain

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F. DELGADO

University of Valladolid, Department of Algebra, Geometry and Topology, 47011 Valladolid, Spain

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S. M. GUSEIN-ZADE

Moscow State University, Faculty of Mathematics and Mechanics, Moscow, GSP-1, 119991, Russia

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, and

F. HERNANDO

University College Cork, Department of Mathematics, Cork, Ireland

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https://doi.org/10.1142/S0129167X10006586Cited by:4 (Source: Crossref)

Abstract

In earlier papers there were given formulae for the Poincaré 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 Poincaré series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincaré series of a general collection of valuations on the ring centered at the origin and prove a generalization of the statement that the Poincaré series determines the topology of the collection.

Keywords:

AMSC: 14H20, 32S45, 13F30

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