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

MathSciNet review:
2888234

Full-text PDF

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.

**[Bv]**Alexander I. Barvinok,*A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed*, Math. Oper. Res.**19**(1994), no. 4, 769–779. MR**1304623**, 10.1287/moor.19.4.769**[BP]**Alexander Barvinok and James E. Pommersheim,*An algorithmic theory of lattice points in polyhedra*, New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97), Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ. Press, Cambridge, 1999, pp. 91–147. MR**1731815****[BV]**Nicole Berline and Michèle Vergne,*Local Euler-Maclaurin formula for polytopes*, Mosc. Math. J.**7**(2007), no. 3, 355–386, 573 (English, with English and Russian summaries). MR**2343137****[Br]**Michel Brion,*Points entiers dans les polyèdres convexes*, Ann. Sci. École Norm. Sup. (4)**21**(1988), no. 4, 653–663 (French). MR**982338****[Fu]**William Fulton,*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037****[La]**Jim Lawrence,*Rational-function-valued valuations on polyhedra*, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc., Providence, RI, 1991, pp. 199–208. MR**1143297****[McM]**P. McMullen,*Lattice invariant valuations on rational polytopes*, Arch. Math. (Basel)**31**(1978/79), no. 5, 509–516. MR**526617**, 10.1007/BF01226481**[Mo]**Robert Morelli,*Pick’s theorem and the Todd class of a toric variety*, Adv. Math.**100**(1993), no. 2, 183–231. MR**1234309**, 10.1006/aima.1993.1033**[Pi]**G.A. Pick,*Geometrisches zur Zahlenlehre*, Sitzenber. Lotos (Prague)**19**(1899) 311-319.**[PT]**James Pommersheim and Hugh Thomas,*Cycles representing the Todd class of a toric variety*, J. Amer. Math. Soc.**17**(2004), no. 4, 983–994. MR**2083474**, 10.1090/S0894-0347-04-00460-6**[Th]**Hugh Thomas,*Cycle-level intersection theory for toric varieties*, Canad. J. Math.**56**(2004), no. 5, 1094–1120. MR**2085635**, 10.4153/CJM-2004-049-0

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
57N10,
57M25

Retrieve articles in all journals with MSC (2010): 57N10, 57M25

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