Abstract
To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.
Similar content being viewed by others
References
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), 47–54 (2003)
Campillo, A., Delgado, F., Gusein-Zade, S.M.: The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors. Funct. Anal. Other Math. 3(1), 39–46 (2010)
Campillo, A., Delgado, F., Gusein-Zade, S.M.: Multi-index filtrations and generalized Poincaré series. Monatsh. Math. 150(3), 193–209 (2007)
Campillo, A., Delgado, F., Gusein-Zade, S.M., Hernando, F.: Poincaré series of collections of plane valuations. Int. J. Math. 21(11), 1461–1473 (2010)
Campillo, A., Delgado, F., Kiyek, K.: Gorenstein property and symmetry for one-dimensional local Cohen–Macaulay rings. Manuscr. Math. 83(3–4), 405–423 (1994)
Cutkosky, S.D., Herzog, J., Reguera, A.: Poincaré series of resolutions of surface singularities. Trans. Am. Math. Soc. 356(5), 1833–1874 (2004)
Delgado, F.: The semigroup of values of a curve singularity with several branches. Manuscr. Math. 59, 347–374 (1987)
Denef, J., Loeser, F.: Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135(1), 201–232 (1999)
Gorsky E., Némethi A. Poincaré series of algebraic links and lattice homology. arXiv: 1301.7636 (2013)
Moyano-Fernández, J.J., Zúñiga-Galindo, W.A.: Motivic zeta functions for curve singularities. Nagoya Math. J. 198, 47–75 (2010)
Spivakovky, M.: Valuations in function fields of surfaces. Am. J. Math 112, 107–156 (1990)
Yamamoto, M.: Classification of isolated algebraic singularities by their Alexander polynomials. Topology 23(3), 277–287 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
Partially supported by the grant MTM2007-64704 and MTM2012-36917-C03-01 / 02 (both grants with the help of FEDER Program). Third author is also partially supported by the Russian government grant 11.G34.31.0005, RFBR–13-01-00755, NSh–4850.2012.1 and Simons-IUM fellowship.
Rights and permissions
About this article
Cite this article
Campillo, A., Delgado, F. & Gusein-Zade, S.M. Hilbert function, generalized Poincaré series and topology of plane valuations. Monatsh Math 174, 403–412 (2014). https://doi.org/10.1007/s00605-013-0547-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-013-0547-5