Exact Euler Maclaurin Formulas for Simple Lattice Polytopes

Euler Maclaurin formulas for a polytope express the sum of the values of a function over the lattice points in the polytope in terms of integrals of the function and its derivatives over faces of the polytope or its dilations. There are two kinds of Euler Maclaurin formulas: exact formulas, which apply to exponential or polynomial functions, and formulas with remainder, which apply to arbitrary smooth functions. Exact Euler Maclaurin formulas were given by Khovanskii and Pukhlikov for regular polytopes and by Cappell-Shaneson, Guillemin, and Brion-Vergne for simple polytopes. Formulas with remainder were given in [KSW1,KSW2] for simple polytopes; these also imply exact formulas. In this paper we give new proofs of the exact formulas for generic exponential functions on simple polytopes, following the lines of the original Khovanskii-Pukhlikov proof. We then use an algebraic formalism due to Cappell and Shaneson to explain the equivalence of the different formulas. Finally, using this formalism, we prove that the exact formulas also hold for polynomial functions.