Abstract
We define a new equivariant (with respect to a finite group \(G\) action) version of the Poincaré series of a multi-index filtration as an element of the power series ring \({\widetilde{A}}(G)[[t_1, \ldots , t_r]]\) for a certain modification \({\widetilde{A}}(G)\) of the Burnside ring of the group \(G\). We give a formula for this Poincaré series of a collection of plane valuations in terms of a \(G\)-resolution of the collection. We show that, for filtrations on the ring of germs of functions in two variables defined by the curve valuations corresponding to the irreducible components of a plane curve singularity defined by a \(G\)-invariant function germ, in the majority of cases this equivariant Poincaré series determines the corresponding equivariant monodromy zeta functions defined earlier.
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Supported by the grants MTM2012-36917-C03-01 / 02 (both grants with the help of FEDER Program).
Supported by the grants RFBR–13-01-00755, NSh–5138.2014.1.
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Campillo, A., Delgado, F. & Gusein-Zade, S.M. An equivariant Poincaré series of filtrations and monodromy zeta functions. Rev Mat Complut 28, 449–467 (2015). https://doi.org/10.1007/s13163-014-0160-8
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DOI: https://doi.org/10.1007/s13163-014-0160-8