Geometric motivic Poincar\'e series of quasi-ordinary singularities

TL;DR

This paper provides an explicit description of the geometric motivic Poincaré series for quasi-ordinary hypersurface singularities using Newton polyhedra and logarithmic jacobian ideals, extending the understanding of arc spaces in algebraic geometry.

Contribution

It offers a new explicit formula for the motivic Poincaré series of quasi-ordinary singularities based on Newton polyhedra and characteristic monomials.

Findings

Explicit description of the motivic Poincaré series for quasi-ordinary singularities.

Connection between the series and Newton polyhedra of logarithmic jacobian ideals.

Extension of rationality results to a broader class of singularities.

Abstract

The geometric motivic Poincar\'e 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 $(S,0)$.