Poincar\'e series for filtrations defined by discrete valuations with arbitrary center

Abstract

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 of these valuations and often one uses several definitions for Poincar\'e series. In this article we show that these definitions can differ when the centers of the valuations are not zero-dimensional, i.e. do not have the maximal ideal as center. We give a unifying definition for Poincar\'e series which also allows filtrations defined by valuations that are all nonzero-dimensional. We then show that this definition satisfies a nice relation between Poincar\'e series for embedded filtrations and Poincar\'e series for the ambient space and we give some application for singularities which are nondegenerate with respect to their Newton polyhedron.