The Ehrhart polynomial of a lattice $n$ -simplex
Authors:
Ricardo Diaz and Sinai Robins
Journal:
Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 1-6
MSC (1991):
Primary 52B20, 52C07, 14D25, 42B10, 11P21, 11F20, 05A15; Secondary 14M25, 11H06
DOI:
https://doi.org/10.1090/S1079-6762-96-00001-7
MathSciNet review:
1405963
Full-text PDF Free Access
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Abstract: The problem of counting the number of lattice points inside a lattice polytope in $\mathbb {R}^{n}$ has been studied from a variety of perspectives, including the recent work of Pommersheim and Kantor-Khovanskii using toric varieties and Cappell-Shaneson using Grothendieck-Riemann-Roch. Here we show that the Ehrhart polynomial of a lattice $n$-simplex has a simple analytical interpretation from the perspective of Fourier Analysis on the $n$-torus. We obtain closed forms in terms of cotangent expansions for the coefficients of the Ehrhart polynomial, that shed additional light on previous descriptions of the Ehrhart polynomial.
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Additional Information
Ricardo Diaz
Affiliation:
Department of Mathematics, University of Northern Colorado, Greeley, Colorado 80639
Email:
rdiaz@bentley.univnorthco.edu
Sinai Robins
Affiliation:
Department of Mathematics, UCSD 9500 Gilman Drive, La Jolla, CA 92093-0112
MR Author ID:
342098
Email:
srobins@ucsd.edu
Keywords:
Lattice polytopes,
Ehrhart polynomials,
Fourier analysis,
Laplace transforms,
cones,
Dedekind sums
Received by editor(s):
August 4, 1995
Received by editor(s) in revised form:
December 1, 1995
Additional Notes:
Research partially supported by NSF Grant #9508965.
Communicated by:
Svetlana Katok
Article copyright:
© Copyright 1996
American Mathematical Society
