Electronic Research Announcements

Author(s): Ricardo Diaz; 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
Retrieve article in: PDF DVI PostScript

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.

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

Ricardo Diaz
Affiliation: Department of Mathematics, University of Northern Colorado, Greeley, Colorado 80639
Email: rdiaz@bentley.univnorthco.edu

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1079-6762

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Next Article

Currently published by the American Institute of Mathematical Sciences as Electronic Research Announcements in Mathematical Sciences


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS