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Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials

Published online by Cambridge University Press:  20 November 2018

Ke-Pao Lin  and
Stephen S.-T. Yau
Ke-Pao Lin
Affiliation:
Department of Information Management, Chang Gung Institute of Technology, 261 Wen-Hwa 1 Road, Kwei-Shan, Tao-Yuan, Taiwan 333-03, Republic of China, e-mail: kplin@cc.cgin.edu.tw
Stephen S.-T. Yau
Affiliation:
Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045, U.S.A., e-mail: yau@uic.edu
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Abstract

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Recently there has been tremendous interest in counting the number of integral points in $n$-dimensional tetrahedra with non-integral vertices due to its applications in primality testing and factoring in number theory and in singularities theory. The purpose of this note is to formulate a conjecture on sharp upper estimate of the number of integral points in $n$-dimensional tetrahedra with non-integral vertices. We show that this conjecture is true for low dimensional cases as well as in the case of homogeneous $n$-dimensional tetrahedra. We also show that the Bernoulli polynomials play a role in this counting.

Keywords

11B7511H0611P2111Y99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 2 , 01 June 2003 , pp. 229 - 241
Copyright
Copyright © Canadian Mathematical Society 2003

References

[Br-Ve1] Brion, M. and Vergne, M., An equivariant Riemann-Roch theorem for complete, simplicial toric varieties. J. Reine Angew.Math. 482 (1997), 67–92.Google Scholar
[Br-Ve2] Brion, M. and Vergne, M., Lattice points in simple polytopes. J. Amer. Math. Soc. 10 (1997), 371–392.Google Scholar
[Ca-Sh] Cappell, S. E. and Shaneson, J. L., Genera of algebraic varieties and counting lattice points. Bull. Amer.Math. Soc. 30 (1994), 62–69.Google Scholar
[Di-Ro] Diaz, R. and Robbin, S., The Ehrhart polynomial of a lattice polytope. Ann. of Math. 135 (1997), 503–518.Google Scholar
[Du] Durfee, A. H., The signature of smoothings of complex surface singularities. Math. Ann. 232 (1978), 85–98.Google Scholar
[Eh] Ehrhart, E., Sur un probleme de geometrie diophantienne lineaire II. J. Reine Angrew Math. 227 (1967), 25–49.Google Scholar
[Ha-Li1] Hardy, G. H. and Littlewood, J. E., Some problems of diophantine approximation. Proc. 5th Int. Congress of Mathematics, (1912), 223–229.Google Scholar
[Ha-Li2] Hardy, G. H. and Littlewood, J. E., The lattice points of a right-angled triangle. Proc. LondonMath. Soc. (2) 20 (1921), 15–36.Google Scholar
[Ha-Li3] Hardy, G. H. and Littlewood, J. E., The lattice points of a right-angled triangle (second memoir). Hamburg Math. Abh. 1 (1922), 212–49.Google Scholar
[Ha-Li4] Hardy, G. H. and Littlewood, J. E., A series of coseconts. Bull. Calcutta Math. Soc. 20 (1930), 251–66.Google Scholar
[Hi-Za] Hirzebruch, F. and Zagier, D., The Atiyah-Singer Index Theorem and Elementary Number Theory. Publish or Perish, Inc., Boston, Massachusetts, 1974.Google Scholar
[Ka-Kh] Kanter, J. M. and Khovanskii, A., Une application du Théoréme de Riemann-Roch combinatoire au polyn.ome d’ Ehrhart des polytopes intier de Rd. C. R. Acad. Sci. Paris I 317 (1993), 501–507.Google Scholar
[Li-Ya1] Lin, K.-P., and Yau, S. S.-T., Sharp upper estimate of geometric genus in terms of Milnor number and multiplicty with application on coordinate free characterization of 3-dimensional homogeneous singularities. (preprint).Google Scholar
[Li-Ya2] Lin, K.-P., Analysis of sharp polynomial upper estimate of number of positive integral points in 4-dimensional tetrahedra. J. Reine Angew.Math. 547 (2002), 191–205.Google Scholar
[Li-Ya3] Lin, K.-P., A sharp upper estimate of the number of integral points in 5-dimensional tetrahedra. J. Number Theory 93 (2002), 207–234.Google Scholar
[Me-Te] Merle, M. and Teissier, B., Conditions d'adjonction d'aprés Du Val. Sèminaire sur les singularities des surfaces (center de Math, de l'Ecole Polytechnique, 1976-1977), Lecture Notes in Math. 777, Springer, Berlin, 1980, 229–245.Google Scholar
[Mo1] Mordell, L. J., Lattice points in a tetrahedron and Dedekind sums. J. Indian Math. 15 (1951), 41–46.Google Scholar
[Mor] Morelli, R., Pick's theorem and the Todd class of tori variety. Adv. in Math. 100 (1993), 183–231.Google Scholar
[Po] Pommersheim, J., Toric varieties, lattice points and Dedekind sums. Math. Ann. 295 (1993), 1–24.Google Scholar
[Sp1] Spencer, D. C., On a Hardy-Littlewood problem of Diophantine approximation. Proc. Cambridge Philos. Soc. XXXV(1939), 527–547.Google Scholar
[Sp2] Spencer, D. C., The lattice points of tetrahedron. J. Math. Phys. (3) XXI(1942), 189–197.Google Scholar
[Xu-Ya1] Xu, Y.-J. and Yau, S. S.-T., Sharp estimate of number of integral points in a tetrahedron. J. Reine Angew.Math. 423 (1992), 199–219.Google Scholar
[Xu-Ya2] Xu, Y.-J. and Yau, S. S.-T., Durfee conjecture and coordinate free characterization of homogeneous singularities. J. Differential Geom. 37 (1993), 375–396.Google Scholar
[Xu-Ya3] Xu, Y.-J. and Yau, S. S.-T., Sharp estimate of numbers of integral points in a 4-dimensional tetrahedron. J. Reine Angew.Math. 473 (1996), 1–23.Google Scholar