Geometric motivic Poincaré series of quasi-ordinary singularities

Abstract

Geometric motivic Poincaré series of a germ at a singular point of complex algebraic variety describes the truncated images of the space of arcs through the singular point. Denef and Loeser proved that it has a rational form. In this paper, the authors study an irreducible germ of quasi-ordinary hypersurface singularities and introduce the notion of logarithmic Jacobian ideals. The main result of this paper is to give the explicit rational form of geometric motivic Poincaré series of such a singularity in terms of the lattice and the Newton polyhedra of the logarithmic Jacobian ideals.