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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
MathSciNet review: 1405963
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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

Sinai Robins
Affiliation: Department of Mathematics, UCSD 9500 Gilman Drive, La Jolla, CA 92093-0112

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