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Arzu Boysal; Michèle Vergne
Paradan’s wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator
(Les formules de saut de Paradan pour les fonctions de partitions, et opérateurs de Khovanski-Pukhlikov)
Annales de l'institut Fourier, 59 no. 5 (2009), p. 1715-1752, doi: 10.5802/aif.2475
Article PDF | Reviews MR 2573189 | Zbl 1186.52006 | 1 citation in Cedram
Class. Math.: 52B20, 14M25
Keywords: Polytopes, toric varieties

Résumé - 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.


[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) no. 4, p. 551-595 Article |  MR 2225674 |  Zbl 1105.52001
[2] W. Baldoni-Silva, J. A. De Loera & M. Vergne, “Counting integer flows in networks”, Found. Comput. Math. 4 (2004) no. 3, p. 277-314 Article |  MR 2078665 |  Zbl 1083.68640
[3] Michel Brion & Michèle Vergne, “Residue formulae, vector partition functions and lattice points in rational polytopes”, J. Amer. Math. Soc. 10 (1997) no. 4, p. 797-833 Article |  MR 1446364 |  Zbl 0926.52016
[4] Wolfgang Dahmen & Charles A. Micchelli, “The number of solutions to linear Diophantine equations and multivariate splines”, Trans. Amer. Math. Soc. 308 (1988) no. 2, p. 509-532 Article |  MR 951619 |  Zbl 0655.10013
[5] C. De Concini & C. Procesi, “Topics in hyperplane arrangements, polytopes and box splines”, To appear (available on the personal web page of C. Procesi)
[6] C. De Concini, C. Procesi & M. Vergne, “Vector partition functions and generalized Dahmen-Miccelli spaces”, arXiv 0805.2907 arXiv
[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) no. 4, p. 188-216  MR 1190788 |  Zbl 0798.52010
[8] P.-E. Paradan, “Jump formulas in Hamiltonian Geometry”, arXiv 0411306 arXiv
[9] András Szenes & Michèle Vergne, “Residue formulae for vector partitions and Euler-MacLaurin sums”, Adv. in Appl. Math. 30 (2003) no. 1-2, p. 295-342, Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001) Article |  MR 1979797 |  Zbl 1067.52014