Geodesic
Integration over the space of functions and Poincar\'e series revisited
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 161-175.

Voir la notice de l'article provenant de la source Math-Net.Ru

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{TM_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 = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {161--175},
     publisher = {mathdoc},
     volume = {302},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2018_302_a6/}
}
TY  - JOUR
AU  - S. M. Gusein-Zade
AU  - F. Delgado
AU  - A. Campillo
TI  - Integration over the space of functions and Poincar\'e series revisited
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2018
SP  - 161
EP  - 175
VL  - 302
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2018_302_a6/
LA  - ru
ID  - TM_2018_302_a6
ER  - 
%0 Journal Article
%A S. M. Gusein-Zade
%A F. Delgado
%A A. Campillo
%T Integration over the space of functions and Poincar\'e series revisited
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2018
%P 161-175
%V 302
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2018_302_a6/
%G ru
%F TM_2018_302_a6
S. M. Gusein-Zade; F. Delgado; A. Campillo. Integration over the space of functions and Poincar\'e series revisited. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Topology and physics, Tome 302 (2018), pp. 161-175. http://geodesic.mathdoc.fr/item/TM_2018_302_a6/

[1] N. Bourbaki, Éléments de mathématique, Part 1: Les structures fondamentales de l'analyse, Ch. III, v. I, Théorie des ensembles. Fascicule de résultats, Hermann, Paris, 1958 | MR

[2] Campillo A., Delgado F., Gusein-Zade S. M., “The Alexander polynomial of a plane curve singularity via the ring of functions on it”, Duke Math. J., 117:1 (2003), 125–156 | DOI | MR

[3] Campillo A., Delgado F., Gusein-Zade S. M., “The Alexander polynomial of a plane curve singularity and integrals with respect to the Euler characteristic”, Int. J. Math., 14:1 (2003), 47–54 | DOI | MR

[4] Campillo A., Delgado F., Gusein-Zade S. M., “Poincaré series of a rational surface singularity”, Invent. math., 155:1 (2004), 41–53 | DOI | MR

[5] Campillo A., Delgado F., Gusein-Zade S. M., “Multi-index filtrations and generalized Poincaré series”, Monatsh. Math., 150:3 (2007), 193–209 | DOI | MR

[6] Campillo A., Delgado F., Gusein-Zade S. M., “Equivariant Poincaré series of filtrations”, Rev. Mat. Complut., 26:1 (2013), 241–251 | DOI | MR

[7] Campillo A., Delgado F., Gusein-Zade S. M., “An equivariant Poincaré series of filtrations and monodromy zeta functions”, Rev. Mat. Complut., 28:2 (2015), 449–467 | DOI | MR

[8] Campillo A., Delgado F., Gusein-Zade S. M., Hernando F., “Poincaré series of collections of plane valuations”, Int. J. Math., 21:11 (2010), 1461–1473 | DOI | MR

[9] Campillo A., Delgado F., Kiyek K., “Gorenstein property and symmetry for one-dimensional local Cohen–Macaulay rings”, Manuscr. math., 83:3–4 (1994), 405–423 | DOI | MR

[10] Campillo A., Galindo C., “On the graded algebra relative to a valuation”, Manuscr. math., 92:2 (1997), 173–189 | DOI | MR

[11] Delgado F., Gusein-Zade S. M., “Poincaré series for several plane divisorial valuations”, Proc. Edinb. Math. Soc. Ser. 2, 46:2 (2003), 501–509 | DOI | MR

[12] Denef J., Loeser F., “Germs of arcs on singular algebraic varieties and motivic integration”, Invent. math., 135:1 (1999), 201–232 | DOI | MR

[13] Dieck T. tom, Transformation groups and representation theory, Lect. Notes Math., 766, Springer, Berlin, 1979 | DOI | MR

[14] S. M. Gusein-Zade, F. Delgado, A. Campillo, “Integration with respect to the Euler characteristic over a function space and the Alexander polynomial of a plane curve singularity”, Russ. Math. Surv., 55:6 (2000), 1148–1149 | DOI | DOI | MR

[15] Gusein-Zade S. M., Luengo I., Melle-Hernández A., “A power structure over the Grothendieck ring of varieties”, Math. Res. Lett., 11:1 (2004), 49–57 | DOI | MR

[16] Viro O.Ya., “Some integral calculus based on Euler characteristic”, Topology and geometry: Rohlin seminar, Lect. Notes Math., 1346, Springer, Berlin, 1988, 127–138 | DOI | MR