The
arithmetic motivic Poincaréseries 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.
Instituto de Ciencias
Matemáticas
Departamento de Álgebra
Facultad de Ciencias Matemáticas
Universidad Complutense de Madrid
Plaza de las Ciencias 3
28040 Madrid
Spain