Poincaré Series of Embedded Filtrations

In this article, we define a Poincaré series on a subspace of a complex analytic germ, induced by a multi-index filtration on the ambient space. This Poincaré series differs from Poincaré series studied before in the sense that there is no notion of fibre that corresponds to our Poincaré series. We compute this Poincaré series for subspaces defined by principal ideals. For plane curve singularities and nondegenerate singularities this Poincaré series yields topological and geometric information. We compare this Poincaré series with the one introduced in [E,G-Z2]. In few cases, they are equal and we show that the Poincaré series we consider in this paper in general yields more information.

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Published 28 October 2011