Moscow Mathematical Journal

Volume 25, Issue 2, April–June 2025 pp. 153–161.

Real Poincaré Series of a Plane Divisorial Valuation

Authors: A. Campillo (1), F. Delgado (2), and S. M. Gusein-Zade (3)
Author institution:(1) IMUVA (Instituto de Investigación en Matemáticas), Universidad de Valladolid, Paseo de Belén, 7, 47011 Valladolid, Spain
(2) IMUVA (Instituto de Investigación en Matemáticas), Universidad de Valladolid, Paseo de Belén, 7, 47011 Valladolid, Spain
(3) Moscow State University, Faculty of Mathematics and Mechanics, Moscow Center for Fundamental and Applied Mathematics, Moscow, Leninskie Gory 1, GSP-1, 119991, Russia &
National Research University “Higher School of Economics”, Usacheva street 6, Moscow, 119048, Russia

Summary:

Earlier, there was computed the Poincaré series of a valuation or of a collection of valuations on the ring of germs of holomorphic functions in two variables. For a collection of several plane curve valuations it appeared to coincide with the Alexander polynomial of the corresponding algebraic link and therefore determines the embedded topology of the the collection of the curves. Recently, the authors defined two versions of the Poincaré series of a valuation or of a collection of valuations in the real setting. These two Poincaré series (and also the so called semigroup Poincaré series, which essentially makes sense only for one valuation) were computed for one plane curve valuation. Here we compute them for a plane divisorial valuation.

2020 Math. Subj. Class. 16W60, 14B05.

Keywords: Poincaré series, germs of real functions, plane divisorial valuation.

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