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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2018, Volume 302, Pages 161–175
DOI: https://doi.org/10.1134/S037196851803007X
(Mi tm3919)
 

Integration over the space of functions and Poincaré series revisited

S. M. Gusein-Zadea, F. Delgadob, A. Campillob

a Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia
b IMUVA (Instituto de Investigación en Matemáticas), Universidad de Valladolid, Paseo de Belén, 7, 47011 Valladolid, Spain
References:
Abstract: 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.
Keywords: filtrations, plane valuations, Poincaré series, integration with respect to the Euler characteristic.
Funding agency Grant number
Russian Science Foundation 16-11-10018
Ministerio de Economía y Competitividad de España MTM2015-65764-C3-1-P
The research of S. M. Gusein-Zade (Sections 1, 3, and 7) was supported by the Russian Science Foundation under grant 16-11-10018. The second and third authors were supported by the grant MTM2015-65764-C3-1-P (with the help of FEDER program).
Received: November 27, 2017
English version:
Proceedings of the Steklov Institute of Mathematics, 2018, Volume 302, Pages 146–160
DOI: https://doi.org/10.1134/S008154381806007X
Bibliographic databases:
Document Type: Article
UDC: 512.717
Language: Russian
Citation: S. M. Gusein-Zade, F. Delgado, A. Campillo, “Integration over the space of functions and Poincaré series revisited”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 161–175; Proc. Steklov Inst. Math., 302 (2018), 146–160
Citation in format AMSBIB
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\paper Integration over the space of functions and Poincar\'e series revisited
\inbook Topology and physics
\bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday
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\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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