  In this article we define a Poincare series on a subspace of a complex
analytic germ, induced by a multi-index filtration on the ambient space. We
compute this Poincare series for subspaces defined by principal ideals. For
plane curve singularities and nondegenerate singularities this Poincare series
yields topological and geometric information. We compare this Poincare series
with the one introduced in [E,G-Z]. In few cases they are equal and we show
that the Poincare series we consider in this paper in general yields more
information.
