Sum-integral Interpolators and the Euler-maclaurin Formula for Polytopes
نویسندگان
چکیده
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-interpolator (which interpolates between integrals and sums over interior lattice points.) 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 and unify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.
منابع مشابه
The Euler-Maclaurin formula for simple integral polytopes.
We give a Euler-Maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope. Our proof uses elementary methods.
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