Poincare series of embedded filtrations

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.