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arXiv · 1002.3522

Sum-integral interpolators and the Euler-Maclaurin formula for polytopes

Stavros Garoufalidis·James E. Pommersheim
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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 \ISo-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 and unify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.

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Stavros Garoufalidis, James E. Pommersheim. 2010-05-20. Sum-integral interpolators and the Euler-Maclaurin formula for polytopes. https://arxiv.org/abs/1002.3522

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