Let $H$ denote the set of formal arcs going through a singular
point of an algebraic variety $V$ defined over an algebraically
closed field $k$ of characteristic zero. In the late sixties,
J. Nash has observed that for any nonnegative integer $s$, the set
$j^s(H)$ of $s$-jets of arcs in $H$ is a constructible subset of
some affine space. Recently (1999), J. Denef and F. Loeser have
proved that the Poincar\'{e} series associated with the image of
$j^s(H)$ in some suitable localization of the Grothendieck ring of
algebraic varieties over $k$ is a rational function. We compute
this function for normal toric surface singularities.
@article{1063050167,
author = {Lejeune-Jalabert, Monique and Reguera, Ana J.},
title = {The Denef-Loeser series for toric surface singularities},
journal = {Rev. Mat. Iberoamericana},
volume = {19},
number = {2},
year = {2003},
pages = { 581-612},
language = {en},
url = {http://dml.mathdoc.fr/item/1063050167}
}
Lejeune-Jalabert, Monique; Reguera, Ana J. The Denef-Loeser series for toric surface singularities. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp. 581-612. http://gdmltest.u-ga.fr/item/1063050167/