Geometric motivic Poincaré series of quasi-ordinary singularities

Abstract

AbstractThegeometric motivic Poincaré seriesof 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 ofquasi-ordinary hypersurface singularityin terms of the Newton polyhedra of thelogarithmic jacobian ideals. These ideals are determined by thecharacteristic monomialsof a quasi-ordinary branch parametrizing (S, 0).