Integration over the Space of Functions and Poincaré Series Revisited

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 b...

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Bibliographic Details
Published in:Proceedings of the Steklov Institute of Mathematics 2018-08, Vol.302 (1), p.146-160
Main Authors: Gusein-Zade, S. M., Delgado, F., Campillo, A.
Format: Article
Language:English
Subjects:
Integrals
Mathematics
Mathematics and Statistics
Software
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Summary: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 Poincar´e 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 Poincar´e series (including equivariant ones) of collections of plane valuations which contain valuations not centred at the origin.
ISSN:0081-5438
1531-8605
DOI:10.1134/S008154381806007X