Geometric motivic Poincaré series of quasi-ordinary singularities

The geometric motivic Poincaré series of a germ (S, 0) of complex algebraic variety takes into account the classes in the Grothendieck ring of the jets of arcs through (S, 0). Denef and Loeser proved that this series has a rational form. We give an explicit description of this invariant when (S, 0) is an irreducible germ of quasi-ordinary hypersurface singularity in terms of the Newton polyhedra of the logarithmic jacobian ideals. These ideals are determined by the characteristic monomials of a quasi-ordinary branch parametrizing (S, 0).