Multi-index Filtrations and Generalized Poincaré Series

Abstract

A multi-index filtration on the ring of germs of functions can be described by its Poincare series. We consider a finer invariant (or rather two invariants) of a multi-index filtration than the Poincare series generalizing the last one. The construction is based on the fact that the Poincare series can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring of functions. The generalization of the Poincare series is defined as a similar integral with respect to the generalized Euler characteristic with values in the Grothendieck ring of varieties. For the filtration defined by orders of functions on the components of a plane curve singularity C and for the so called divisorial filtration for a modification of $({\Bbb C}^2,0)$ by a sequence of blowing-ups there are given formulae for this generalized Poincare series in terms of an embedded resolution of the germ C or in terms of the modification respectively. The generalized Euler characteristic of the extended semigroup corresponding to the divisorial filtration is computed giving a curious “motivic version” of an A’Campo type formula.