Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Residue formulae, vector partition functions
and lattice points in rational polytopes

Authors: Michel Brion and Michèle Vergne
Journal: J. Amer. Math. Soc. 10 (1997), 797-833
MSC (1991): Primary 11P21, 52B20
MathSciNet review: 1446364
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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.

References [Enhancements On Off] (What's this?)

  • [1] M. F. Atiyah, Elliptic operators and compact groups, Springer-Verlag, New York, 1974. MR 58:2910
  • [2] A. I. Barvinok, Computing the volume, counting integral points, and exponential sums, Discrete Comput. Geom. 10 (1993), 123-141. MR 94d:52005
  • [3] M. Brion and M. Vergne, Lattice points in simple polytopes, J. Amer. Math. Soc. 10 (1997), 371-392. CMP 97:06
  • [4] M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. reine angew. Math. 482 (1997), 67-92. CMP 97:06
  • [5] M. Brion et M. Vergne, Une formule d'Euler-MacLaurin pour les fonctions de partition, C. R. Acad. Sci. Paris (Série I) 322 (1996), 217-220. MR 97a:11164
  • [6] M. Brion et M. Vergne, Une formule d'Euler-MacLaurin pour les polytopes convexes rationnels, C. R. Acad. Sci. Paris (Série I) 322 (1996), 317-320. MR 97a:11165
  • [7] S. E. Cappell and J. L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. 30 (1994), 62-69. MR 94f:14018
  • [8] S. E. Cappell and J. L. Shaneson, Euler-MacLaurin expansions for lattices above dimension one, C. R. Acad. Sci. Paris (Série I) 321 (1995), 885-890. MR 96i:52012
  • [9] R. Diaz and S. Robins, The Ehrhart polynomial of a lattice $n$-simplex, Electronic Research Announcements of the AMS 2 (1996). CMP 96:17
  • [10] E. Ehrhart, Polyèdres et réseaux, J. reine angew. Math. 226 (1967), 1-29. MR 35:4184
  • [11] V. Ginzburg, V. Guillemin and Y. Karshon, Cobordism techniques in symplectic geometry, The Carus Mathematical Monographs,, Mathematical Association of America, to appear.
  • [12] V. Guillemin, Riemann-Roch for toric orbifolds, preprint (1995).
  • [13] M-N. Ishida, Polyhedral Laurent series and Brion's equalities, International J. Math. 1 (1990), 251-265. MR 91m:14081
  • [14] L. C. Jeffrey and F. C. Kirwan, Localization for non-abelian group actions, Topology 34 (1995), 291-327. CMP 95:08
  • [15] J-M. Kantor and A. Khovanskii, Une application du théorème de Riemann-Roch combinatoire au polynôme d'Ehrhart des polytopes entiers de ${\mathbb {R}}^{d}$, C. R. Acad. Sci. Paris (Série I) 317 (1993), 501-507. MR 94k:52018
  • [16] T. Kawasaki, The Riemann-Roch theorem for complex $V$-manifolds, Osaka J. Math. 16 (1979), 151-159. MR 80f:58042
  • [17] A. Khovanskii and A. Pukhlikov, A Riemann-Roch theorem for integrals and sums of quasipolynomials over virtual polytopes, St-Petersburg Math. J. 4 (1993), 789-812. MR 94c:14044
  • [18] P. McMullen, Transforms, diagrams and representations, Proc. Geometry Symposium, Siegen 1978, Birkhäuser, Basel, 1979, pp. 92-130. MR 81i:52007
  • [19] R. Morelli, A Theory of Polyhedra, Adv. Math. 9 (1993), 1-73. MR 94f:52023
  • [20] B. Sturmfels, On vector partition functions, J. Combinatorial Theory, Series A 72 (1995), 302-309. MR 97b:52014
  • [21] M. Vergne, Equivariant index formulas for orbifolds, Duke Math. J. 82 (1996), 637-652. CMP 96:12

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 11P21, 52B20

Retrieve articles in all journals with MSC (1991): 11P21, 52B20

Additional Information

Michel Brion
Affiliation: Institut Fourier, B.P. 74, 38402 Saint-Martin d’Hères Cedex, France

Michèle Vergne
Affiliation: École Normale Supérieure, 45 rue d’Ulm, 75005 Paris Cedex 05, France

Keywords: Vector partition functions, rational convex polytopes
Received by editor(s): December 30, 1996
Received by editor(s) in revised form: March 28, 1997
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society