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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
DOI: https://doi.org/10.1090/S1079-6762-96-00001-7
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.


References [Enhancements On Off] (What's this?)

  • 1. M. Brion, Points entiere dans les polyèdres convexes, Ann. Sci. Ecole Nor. Sup. ($4^{c}$) 21 (1988), 653--663. MR 90d:52020
  • 2. S. E. Cappell and J. L. Shaneson, Genera of algebraic varieties and counting of lattice points, Bulletin of the AMS (Jan. 1994), 62--69. MR 94f:14018
  • 3. V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33 (1978), 97--154. MR 80g:14001
  • 4. E. Ehrhart, Sur un problème de géométrie diophantienne linéaire II, J. Reine Angew. Math. 227 (1967), 25--49. MR 36:105
  • 5. F. Hirzebruch and D. Zagier, The Atiyah-Singer index theorem and elementary number theory, Publish or Perish Press, Boston, MA, 1974. MR 58:31291
  • 6. J. M. Kantor and A. Khovanskii, Une application du Théorème de Riemann-Roch combinatoire au polynôme d'Ehrhart des polytopes entier de $\mathbb {R}^{d}$, C. R. Acad. Sci. Paris, Series I 317 (1993), 501--507. MR 94k:52018
  • 7. I. G. MacDonald, Polynomials associated with finite cell complexes, J. London Math. Soc. (2) 4 (1971), 181--192. MR 45:7594
  • 8. L. J. Mordell, Lattice points in a tetrahedron and generalized Dedekind sums, J. Indian Math. 15 (1951), 41--46. MR 13:322b
  • 9. M. Newman, Integral matrices, Academic Press, New York, 1972. MR 49:5038
  • 10. G. Pick, Geometrisches zur Zahlenlehre, Sitzungber. Lotos (Prague) 19 (1899), 311--319.
  • 11. M. A. Pinsky, Pointwise Fourier inversion in several variables, Notices of the AMS 42 (1995), 330--334. MR 96b:42010
  • 12. J. Pommersheim, Toric varieties, lattice points, and Dedekind sums, Math. Annalen 295, 1 (1993), 1--24. MR 94c:14043
  • 13. B. Randol, On the Fourier transform of the indicator function of a planar set, Trans. of the AMS (1969), 271--278. MR 40:4678a
  • 14. C. L. Siegel, Über Gitterpunkte in konvexen Körpern und ein damit zusammenhängendes Extremalproblem, Acta Math. 65 (1935), 307--323.
  • 15. E. Stein and G. Weiss, Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, NJ, 1971. MR 46:4102
  • 16. R. Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194--253. MR 54:111
  • 17. D. Gabai, Genera of the arborescent links, and W. Thurston, A norm for the homology of 3-manifolds, Memoirs of the AMS 59 (1986). MR 87h:57010; MR 88h:57014
  • 18. Y.-J. Xu and S.S.-T. Yau, A sharp estimate of the number of integral points in a tetrahedron, J. Reine Angew. Math. 423 (1992), 119--219. MR 93d:11067
  • 19. D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973), 149--172. MR 50:9801

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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
Email: srobins@ucsd.edu

DOI: https://doi.org/10.1090/S1079-6762-96-00001-7
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

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