Poincaré series and zeta function of the monodromy of a quasihomogeneous singularity.
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 \(m\).
scientific article; zbMATH DE number 1886045
scientific article