Abstract
To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.
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Communicated by A. Constantin.
Partially supported by the grant MTM2007-64704 and MTM2012-36917-C03-01 / 02 (both grants with the help of FEDER Program). Third author is also partially supported by the Russian government grant 11.G34.31.0005, RFBR–13-01-00755, NSh–4850.2012.1 and Simons-IUM fellowship.
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Campillo, A., Delgado, F. & Gusein-Zade, S.M. Hilbert function, generalized Poincaré series and topology of plane valuations. Monatsh Math 174, 403–412 (2014). https://doi.org/10.1007/s00605-013-0547-5
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DOI: https://doi.org/10.1007/s00605-013-0547-5