Geodesic
Alexander polynomials and Poincar\'e series of sets of ideals
Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 4, pp. 40-48

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Earlier the authors considered and, in some cases, computed Poincaré series for two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular, on the complex plane). A filtration of the first class was defined by a curve (with several branches) on the surface singularity. A filtration of the second class (called divisorial) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define and compute in some cases the Poincaré series corresponding to a set of ideals in the ring of germs of functions on a surface singularity. For the complex plane, this notion unites the two classes of filtrations described above.
Keywords: ideal, Poincaré series, zeta function.
Mots-clés : surface 
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     author = {S. M. Gusein-Zade and F. Delgado and A. Campillo},
     title = {Alexander polynomials and {Poincar\'e} series of sets of ideals},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {40--48},
     publisher = {mathdoc},
     volume = {45},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2011_45_4_a3/}
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S. M. Gusein-Zade; F. Delgado; A. Campillo. Alexander polynomials and Poincar\'e series of sets of ideals. Funkcionalʹnyj analiz i ego priloženiâ, Tome 45 (2011) no. 4, pp. 40-48. http://geodesic.mathdoc.fr/item/FAA_2011_45_4_a3/