Poincaré series of embedded filtrations (Q2428816)

Poincaré series induced by filtrations on the local ring of a complex analytic germ are known to provide important information on the topology of the respective singularity. They have also been intensively studied in the context of the zeta-function of the monodromy. More recently, generalizations of this notion, which arise from certain valuations on the ambient smooth space of the singularity, have been explored. This article introduces another generalization of a similar flavour: a Poincaré series on a subspace of a complex analytic germ induced by a multi-index filtration on the ambient space. This Poincaré series is then studied and compared to previously known notions. Special attention is payed to the cases of plane curve singularities and to embedded filtrations induced from the Newton polyhedron. In the latter case, the Newton polyhedron and the zeta function of the monodromy are shown to be determined by this Poincaré series.