On real analogues of the Poincaré series

2023/06/16 by A. Campillo, Campillo, A., F. Delgado +3
Mathematics · #14B05 #14P15 #16W60 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2306.10148

openalex publication_date 2023/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There exist several equivalent equations for the Poincaré series of a collection of valuations on the ring of germs of functions on a complex analytic variety. We give definitions of the Poincaé series of a collection of valuations in the real setting (i.e., on the ring of germs of functions on a real analytic variety), compute them for the case of one curve valuation on the plane and discuss some of their properties.

Related

• Real Poincaré series of a plane divisorial valuation
• Poincaré series of valuations on functions over a field
• The Poincare series of divisorial valuations in the plane defines the topology of the set of divisors
• Hilbert function, generalized Poincaré series and topology of plane valuations
• Arcs and resolution of singularities