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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Stavros Garoufalidis and James Pommersheim
Journal: Trans. Amer. Math. Soc. 364 (2012), 2933-2958
MSC (2010): Primary 57N10; Secondary 57M25
Published electronically: February 14, 2012
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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 $ \operatorname {SI}$-interpolator (and a local Euler-Maclaurin formula), an $ \operatorname {IS}$-interpolator (and a reverse local Euler-Maclaurin formula) and an $ \operatorname {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.


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Additional Information

Stavros Garoufalidis
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: stavros@math.gatech.edu

James Pommersheim
Affiliation: Department of Mathematics, Reed College, 3203 SE Woodstock Boulevard, Portland, Oregon 97202-8199
Email: jamie@reed.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05381-5
PII: S 0002-9947(2012)05381-5
Keywords: Polytopes, Euler-Maclaurin formula, reverse Euler-Maclaurin formula, lattice points, flag varieties, exponential sums, exponential integrals, interpolators.
Received by editor(s): February 18, 2010
Received by editor(s) in revised form: May 20, 2010
Published electronically: February 14, 2012
Additional Notes: The first author was supported in part by NSF
Article copyright: © Copyright 2012 American Mathematical Society