The arithmetic motivic Poincaré series of a variety 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 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 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.
Helena Cobo Pablos. Pedro Daniel González Pérez. "Arithmetic motivic Poincaré series of toric varieties." Algebra Number Theory 7 (2) 405 - 430, 2013. https://doi.org/10.2140/ant.2013.7.405