Geodesic
Residue formulae, vector partition functions and lattice points in rational polytopes
Brion, Michel1 ; Vergne, Michèle2
1 Institut Fourier, B.P. 74, 38402 Saint-Martin d’Hères Cedex, France
2 École Normale Supérieure, 45 rue d’Ulm, 75005 Paris Cedex 05, France
Journal of the American Mathematical Society, Tome 10 (1997) no. 4, pp. 797-833

Voir la notice de l'article provenant de la source American Mathematical Society

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.
DOI : 10.1090/S0894-0347-97-00242-7

Brion, Michel 1 ; Vergne, Michèle 2

1 Institut Fourier, B.P. 74, 38402 Saint-Martin d’Hères Cedex, France
2 École Normale Supérieure, 45 rue d’Ulm, 75005 Paris Cedex 05, France 
@article{10_1090_S0894_0347_97_00242_7,
     author = {Brion, Michel and Vergne, Mich\~A{\textasciidieresis}le},
     title = {Residue formulae, vector partition functions and lattice points in rational polytopes},
     journal = {Journal of the American Mathematical Society},
     pages = {797--833},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {1997},
     doi = {10.1090/S0894-0347-97-00242-7},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-97-00242-7/}
}
TY  - JOUR
AU  - Brion, Michel
AU  - Vergne, Michèle
TI  - Residue formulae, vector partition functions and lattice points in rational polytopes
JO  - Journal of the American Mathematical Society
PY  - 1997
SP  - 797
EP  - 833
VL  - 10
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-97-00242-7/
DO  - 10.1090/S0894-0347-97-00242-7
ID  - 10_1090_S0894_0347_97_00242_7
ER  - 
%0 Journal Article
%A Brion, Michel
%A Vergne, Michèle
%T Residue formulae, vector partition functions and lattice points in rational polytopes
%J Journal of the American Mathematical Society
%D 1997
%P 797-833
%V 10
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-97-00242-7/
%R 10.1090/S0894-0347-97-00242-7
%F 10_1090_S0894_0347_97_00242_7
Brion, Michel; Vergne, Michèle. Residue formulae, vector partition functions and lattice points in rational polytopes. Journal of the American Mathematical Society, Tome 10 (1997) no. 4, pp. 797-833. doi: 10.1090/S0894-0347-97-00242-7

[1] Atiyah, Michael Francis Elliptic operators and compact groups 1974

[2] Barvinok, Alexander I. Computing the volume, counting integral points, and exponential sums Discrete Comput. Geom. 1993 123 141

[3] Brion, Michel, Vergne, Michã¨Le Une formule d’Euler-Maclaurin pour les fonctions de partition C. R. Acad. Sci. Paris Sér. I Math. 1996 217 220

[4] Brion, Michel, Vergne, Michã¨Le Une formule d’Euler-Maclaurin pour les polytopes convexes rationnels C. R. Acad. Sci. Paris Sér. I Math. 1996 317 320

[5] Cappell, Sylvain E., Shaneson, Julius L. Genera of algebraic varieties and counting of lattice points Bull. Amer. Math. Soc. (N.S.) 1994 62 69

[6] Cappell, Sylvain E., Shaneson, Julius L. Euler-Maclaurin expansions for lattices above dimension one C. R. Acad. Sci. Paris Sér. I Math. 1995 885 890

[7] Ehrhart, E. Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux J. Reine Angew. Math. 1967 1 29

[8] Ishida, Masa-Nori Polyhedral Laurent series and Brion’s equalities Internat. J. Math. 1990 251 265

[9] Kantor, Jean-Michel, Khovanskii, Askold Une application du théorème de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entiers de 𝑅^{𝑑} C. R. Acad. Sci. Paris Sér. I Math. 1993 501 507

[10] Kawasaki, Tetsuro The Riemann-Roch theorem for complex 𝑉-manifolds Osaka Math. J. 1979 151 159

[11] Pukhlikov, A. V., Khovanskiä­, A. G. The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes Algebra i Analiz 1992 188 216

[12] Mcmullen, P. Transforms, diagrams and representations 1979 92 130

[13] Morelli, Robert A theory of polyhedra Adv. Math. 1993 1 73

[14] Sturmfels, Bernd On vector partition functions J. Combin. Theory Ser. A 1995 302 309

Cité par Sources :