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Campillo López, Antonio Delgado de la Mata, Félix Gusein-Zade, Sabir M. Producción Científica 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 permitted to rewrite in new terms known definitions and statements and also appeared to be an effective tool to compute 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 application of this concept for definitions and computations of the Poincar´e series of collections of plane valuations which include valuations not centred at the origin, including equivariant ones. 2019-05-06 2019-05-06 2018 info:eu-repo/semantics/article Proceedings of the Steklov Institute of Mathematics, 2018, vol. 302, p. 1-15 1531-8605 http://uvadoc.uva.es/handle/10324/35937 10.1134/S008154381806007X eng https://link.springer.com/article/10.1134/S008154381806007X#Bib1 info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/4.0/ © 2018 Springer Attribution-NonCommercial-NoDerivatives 4.0 International Springer