Geodesic
Integration over the space of functions and Poincar\'e series revisited
Informatics and Automation, Topology and physics, Tome 302 (2018), pp. 161-175

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Earlier (2000) the authors introduced the notion of the integral with respect to the Euler characteristic over the space of germs of functions on a variety and over its projectivization. This notion allowed the authors to rewrite known definitions and statements in new terms and also turned out to be an effective tool for computing the Poincaré series of multi-index filtrations in some situations. However, the “classical” (initial) notion can be applied only to multi-index filtrations defined by so-called finitely determined valuations (or order functions). Here we introduce a modified version of the notion of the integral with respect to the Euler characteristic over the projectivization of the space of function germs. This version can be applied in a number of settings where the “classical approach” does not work. We give examples of the application of this concept to definitions and computations of the Poincaré series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.
Mots-clés : filtrations, plane valuations
Keywords: Poincaré series, integration with respect to the Euler characteristic. 
@article{TRSPY_2018_302_a6,
     author = {S. M. Gusein-Zade and F. Delgado and A. Campillo},
     title = {Integration over the space of functions and {Poincar\'e} series revisited},
     journal = {Informatics and Automation},
     pages = {161--175},
     publisher = {mathdoc},
     volume = {302},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2018_302_a6/}
}
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S. M. Gusein-Zade; F. Delgado; A. Campillo. Integration over the space of functions and Poincar\'e series revisited. Informatics and Automation, Topology and physics, Tome 302 (2018), pp. 161-175. http://geodesic.mathdoc.fr/item/TRSPY_2018_302_a6/