Quasi Ordinary Singularities, Essential Divisors and Poincare Series

We define Poincaré series associated to a toric or analytically irreducible quasiordinary hypersurface singularity, (S, 0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multigraded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincaré series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincaré series associated to the set of divisorial valuations corresponding to 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 Poincaré series determines and it 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.