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 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.

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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  1. M. Welleda Baldoni, Matthias Beck, Charles Cochet, Michèle Vergne, Volume computation for polytopes and partition functions for classical root systems, Discrete Comput. Geom. 35 (2006), 551-595 Zbl1105.52001MR2225674
  2. W. Baldoni-Silva, J. A. De Loera, M. Vergne, Counting integer flows in networks, Found. Comput. Math. 4 (2004), 277-314 Zbl1083.68640MR2078665
  3. Michel Brion, Michèle Vergne, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc. 10 (1997), 797-833 Zbl0926.52016MR1446364
  4. Wolfgang Dahmen, Charles A. Micchelli, The number of solutions to linear Diophantine equations and multivariate splines, Trans. Amer. Math. Soc. 308 (1988), 509-532 Zbl0655.10013MR951619
  5. C. De Concini, C. Procesi, Topics in hyperplane arrangements, polytopes and box splines Zbl1217.14001
  6. C. De Concini, C. Procesi, M. Vergne, Vector partition functions and generalized Dahmen-Miccelli spaces Zbl1223.58015
  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
  9. András Szenes, Michèle Vergne, Residue formulae for vector partitions and Euler-MacLaurin sums, Adv. in Appl. Math. 30 (2003), 295-342 Zbl1067.52014MR1979797

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