Arithmetic Motivic Poincar\'e series of toric varieties

TL;DR

This paper investigates the arithmetic motivic Poincaré series of affine toric varieties, providing a formula for its rational form based on Newton polyhedra, and identifying potential poles of the series.

Contribution

It offers an explicit formula for the motivic Poincaré series of toric varieties in terms of Newton polyhedra, advancing understanding of this invariant.

Findings

Derived a formula for the series using Newton polyhedra.

Explicitly identified candidate poles of the series.

Extended the study of the invariant to affine toric varieties.

Abstract

The arithmetic motivic Poincar\'e series of a variety $V$ defined over a field of characteristic zero, is an invariant of singularities which was introduced by Denef and Loeser by analogy with the Serre-Oesterl\'e series in arithmetic geometry. They proved that this motivic series has a rational form which specializes to the Serre-Oesterl\'e 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 deduce explicitly a finite set of candidate poles for this invariant.