Stavros Garoufalidis and James Pommersheim
Abstract: A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space V, namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in V. The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in V gives rise to an effectively computable SI-interpolator (and a local Euler-MacLaurin formula), an IS-interpolator (and a reverse local Euler-MacLaurin formula) and an IS^0-interpolator. Rigid complement maps can be constructed by choosing an inner product on V or by choosing a complete flag in V. The corresponding interpolators generalize the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.
Key words: polytopes, Euler-MacLaurin formula, reverse Euler-MacLaurin formula, lattice points, flag varieties, exponential sums, exponential integrals, interpolators.
Notes: 22 pages, 8 figures.