On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions
type: Item
instance of: scholarly article
title: On the monodromy of a plane curve singularity and the Poincaré series of its ring of functions
zbMATH Open document ID: 0967.14017
DOI: 10.1007/BF02465144
publication date: 2001-08-23T00:00:00Z
review text: Let \(f:(\mathbb{C}^2 , 0) \rightarrow (\mathbb{C}, 0)\) be the germ of a holomorphic function with an isolated critical point at the origin and let \((C, 0)\) be the irreducible plane curve singularity defined by \(f = 0\). Let \(V_{f}\) be the Milnor fiber of the singularity and let \(h_{f}: V_{f} \rightarrow V_{f}\) be the monodromy transformation. Let \(\zeta_{f}(t)\) be the zeta function of \(h_{f}\), as defined by \textit{N. A'Campo} [Comment. Math. Helv. 50, 233-248 (1975; Zbl 0333.14008)]. In this paper the authors prove that \(\zeta_{f}(t)\) is exactly the Poincaré series of the ring of functions of \(C\).
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 1407615
zbMATH Keywords: plane curve singularity
zbMATH Keywords: Poincaré series
zbMATH Keywords: zeta function
zbMATH Keywords: critical point
zbMATH Keywords: Milnor fiber
author: Sabir M. Gusein-Zade
author: F. Delgado
author: Antonio Campillo
reviewed by: Cicero Carvalho
MaRDI profile type: Publication
cites work: La fonction zeta d'une monodromie
cites work: Singularities of differentiable maps, Volume 2. Monodromy and asymptotics of integrals. Transl. from the Russian by Hugh Porteous and revised by the authors and James Montaldi
cites work: Algebroid curves in positive characteristic
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1407615