Quasi-ordinary singularities, essential divisors and Poincaré series

Abstract

We define Poincaré series associated to a germ (S, 0) of toric or analytically irreducible quasiordinary hypersurface singularity, by a finite sequence of monomial valuations such that at least one of them is centered at the point 0. This involves the definition of a multi-graded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincar´e series is a rational function with integer coefficients, which can also be defined as an integral with respect to the Euler characteristic of a function defined by the valuations, over the projectivization of the analytic algebra of the singularity. In particular, the Poincaré series associated to the set of divisorial valuations of the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincar´e series determines and is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.