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

DOI:
https://doi.org/10.1090/S0002-9947-2012-05381-5

Published electronically:
February 14, 2012

MathSciNet review:
2888234

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Abstract | References | Similar Articles | Additional Information

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 , namely the family of exponential sums and the family of exponential integrals parametrized by the set of rational polytopes in . The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in gives rise to an effectively computable -interpolator (and a local Euler-Maclaurin formula), an -interpolator (and a reverse local Euler-Maclaurin formula) and an -interpolator (which interpolates between integrals and sums over interior lattice points). Rigid complement maps can be constructed by choosing an inner product on or by choosing a complete flag in . 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:
https://doi.org/10.1090/S0002-9947-2012-05381-5

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