Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator

Arzu Boysal[1]; Michèle Vergne[2]

  • [1] Bogaziçi University Faculty of Arts and Science Department of Mathematics 34342, Bebek-Istanbul (Turkey)
  • [2] Ecole Polytechnique Centre de Mathématiques Laurent Schwartz 91128 Palaiseau Cedex (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 5, page 1715-1752
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Boysal, Arzu, and Vergne, Michèle. "Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator." Annales de l’institut Fourier 59.5 (2009): 1715-1752. <http://eudml.org/doc/10439>.

@article{Boysal2009,
abstract = {Let $P(s)$ be a family of rational polytopes parametrized by inequations. It is known that the volume of $P(s)$ is a locally polynomial function of the parameters. Similarly, the number of integral points in $P(s)$ 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.},
affiliation = {Bogaziçi University Faculty of Arts and Science Department of Mathematics 34342, Bebek-Istanbul (Turkey); Ecole Polytechnique Centre de Mathématiques Laurent Schwartz 91128 Palaiseau Cedex (France)},
author = {Boysal, Arzu, Vergne, Michèle},
journal = {Annales de l’institut Fourier},
keywords = {Polytopes; toric varieties; polytopes},
language = {eng},
number = {5},
pages = {1715-1752},
publisher = {Association des Annales de l’institut Fourier},
title = {Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator},
url = {http://eudml.org/doc/10439},
volume = {59},
year = {2009},
}

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
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 5
SP - 1715
EP - 1752
AB - Let $P(s)$ be a family of rational polytopes parametrized by inequations. It is known that the volume of $P(s)$ is a locally polynomial function of the parameters. Similarly, the number of integral points in $P(s)$ 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.
LA - eng
KW - Polytopes; toric varieties; polytopes
UR - http://eudml.org/doc/10439
ER -

References

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  7. A. G. Khovanskiĭ, A. V. Pukhlikov, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes, Algebra i Analiz 4 (1992), 188-216 Zbl0798.52010MR1190788
  8. P.-E. Paradan, Jump formulas in Hamiltonian Geometry Zbl1253.53083
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