[Les formules de saut de Paradan pour les fonctions de partitions, et opérateurs de Khovanski-Pukhlikov]
Soit une famille de polytopes rationnels paramétrés par des inéquations. On sait que le volume de est une fonction localement polynomiale des paramètres. Similairement, le nombre de points entiers dans est une fonction localement quasi-polynomiale des paramètres. Paul-Émile Paradan a donné une formule de saut pour cette fonction, lorsqu’on traverse un mur. Dans cet article, nous donnons une démonstration algébrique de ces formules de saut. Nous exprimons aussi le saut, à l’aide d’une formule de résidus, ce qui permet de le calculer.
Let be a family of rational polytopes parametrized by inequations. It is known that the volume of is a locally polynomial function of the parameters. Similarly, the number of integral points in is a locally quasi-polynomial function of the parameters. Paul-Émile Paradan proved a jump formula for this function, when crossing a wall. In this article, we give an algebraic proof of this formula. Furthermore, we give a residue formula for the jump, which enables us to compute it.
Keywords: Polytopes, toric varieties
Mot clés : polytopes, variétés toriques
@article{AIF_2009__59_5_1715_0,
author = {Boysal, Arzu and Vergne, Mich\`ele},
title = {Paradan{\textquoteright}s wall crossing formula for partition functions and {Khovanski-Pukhlikov} differential operator},
journal = {Annales de l'Institut Fourier},
pages = {1715--1752},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {59},
number = {5},
year = {2009},
doi = {10.5802/aif.2475},
zbl = {1186.52006},
mrnumber = {2573189},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2475/}
}
TY - JOUR AU - Boysal, Arzu AU - Vergne, Michèle TI - Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator JO - Annales de l'Institut Fourier PY - 2009 SP - 1715 EP - 1752 VL - 59 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2475/ DO - 10.5802/aif.2475 LA - en ID - AIF_2009__59_5_1715_0 ER -
%0 Journal Article %A Boysal, Arzu %A Vergne, Michèle %T Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator %J Annales de l'Institut Fourier %D 2009 %P 1715-1752 %V 59 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2475/ %R 10.5802/aif.2475 %G en %F AIF_2009__59_5_1715_0
Boysal, Arzu; Vergne, Michèle. Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator. Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1715-1752. doi : 10.5802/aif.2475. http://www.numdam.org/articles/10.5802/aif.2475/
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