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2013 Arithmetic motivic Poincaré series of toric varieties
Helena Cobo Pablos, Pedro Daniel González Pérez
Algebra Number Theory 7(2): 405-430 (2013). DOI: 10.2140/ant.2013.7.405


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.


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Helena Cobo Pablos. Pedro Daniel González Pérez. "Arithmetic motivic Poincaré series of toric varieties." Algebra Number Theory 7 (2) 405 - 430, 2013.


Received: 7 November 2011; Revised: 31 January 2012; Accepted: 3 March 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1279.14066
MathSciNet: MR3123644
Digital Object Identifier: 10.2140/ant.2013.7.405

Primary: 14M25
Secondary: 14B05 , 14J17

Keywords: arc spaces , arithmetic motivic Poincaré series , singularities , toric geometry

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 2 • 2013
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