Arithmetic motivic Poincaré series of toric varieties

Abstract

The arithmetic motivic Poincaré series of a variety $V$ defined over a field of characteristic zero is an invariant of singularities that was introduced by Denef and Loeser by analogy with the Serre–Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form that specializes to the Serre–Oesterlé series when $V$ is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper, we study this motivic series when $V$ is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we explicitly deduce a finite set of candidate poles for this invariant.