Residue Formulae, Vector Partition Functions and Lattice Points in Rational Polytopes

We obtain residue formulae for certain functions of several vari-ables. As an application, we obtain closed formulae for vector partition func-tions and for their continuous analogs. They imply an Euler-MacLaurin sum-mation formula for vector partition functions, and for rational convex poly-topes as well: we express the sum of values of a polynomial function at alllattice points of a rational convex polytope in terms of the variation of theintegral of the function over the deformed polytope.Institut Fourier, B.P. 74, 38402 Saint-Martin d’Hères Cedex, FranceE-mail address: [email protected] École Normale Supérieure, 45 rue d’Ulm, 75005 Paris Cedex 05, FranceE-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use