Sumintegral interpolators and the EulerMaclaurin formula for polytopes
Authors:
Stavros Garoufalidis and James Pommersheim
Journal:
Trans. Amer. Math. Soc. 364 (2012), 29332958
MSC (2010):
Primary 57N10; Secondary 57M25
Published electronically:
February 14, 2012
MathSciNet review:
2888234
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Abstract: A local lattice point counting formula, and more generally a local EulerMaclaurin 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 EulerMaclaurin formula), an interpolator (and a reverse local EulerMaclaurin 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 BerlineVergne, PommersheimThomas, and Morelli.
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Additional Information
Stavros Garoufalidis
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 303320160
Email:
stavros@math.gatech.edu
James Pommersheim
Affiliation:
Department of Mathematics, Reed College, 3203 SE Woodstock Boulevard, Portland, Oregon 972028199
Email:
jamie@reed.edu
DOI:
http://dx.doi.org/10.1090/S000299472012053815
Keywords:
Polytopes,
EulerMaclaurin formula,
reverse EulerMaclaurin 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
