Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity.
type: Item
instance of: scholarly article
title: Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity.
zbMATH Open document ID: 1056.14003
published in: Mathematical Research Letters
publication date: 2002-01-01T00:00:00Z
review text: Let \(f:\mathbb C^n \rightarrow \mathbb C\) be a quasihomogeneous polynomial of degree \(d\), non-degenerated with respect to its Newton diagram in the sense of Varchenko, and \(P(t) = \sum_s \dim(A_s)t^s\) be the Poincaré series of the graded ring \(A = \mathbb C[x]/f = \bigoplus_s A_s\). Let \(\zeta^*_f(t)\) be the Saito dual of the zeta function of the monodromy of the singularity defined by \(f\). It is proved that \[ P(t) = \zeta^*_f(t) \prod_m(1 - t^m)^{-\chi(Y_m)}. \] \(\chi(Y_m)\) is the Euler characteristic of \(Y_m\), the set of orbits of the \(\mathbb C^*\)-action on \(V(f) \setminus \{0\}\) for which the isotropy group is the cyclic group of order \(m\).
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S40
zbMATH DE Number: 1886045
author: Wolfgang Ebeling
author: Sabir M. Gusein-Zade
MaRDI profile type: Publication
DOI: 10.4310/MRL.2002.V9.N4.A10
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description: scientific article; zbMATH DE number 1886045