Analysis of sharp polynomial upper estimate of number of positive integral points in a five-dimensional tetrahedra. (English) Zbl 1217.11088
Authors’ summary: Let \(a\geq b\geq c\geq d\geq e\geq 1\) be real numbers and \(P_5\) be the number of positive integral solutions of
\[
\frac xa+\frac yb+\frac zc+\frac ud+\frac ve\leq 1.
\]
In this paper the authors show that \(120 P_5\leq (a-1)(b-1)(c-1)(d-1)(e-1)\). This confirms a conjecture of Durfee for the dimension 5 case. It is also shown that the upper estimate of \(P_5\) given by K.-P. Lin and S. S.-T. Yau [J. Number Theory 93, No. 2, 207–234 (2002; Zbl 0992.11057)] is strictly sharper than that suggested by the Durfee conjecture if \(e\geq \frac{29+\sqrt{489}}{12}\), but is not sharper than that suggested by the Durfee conjecture if \(4\leq e<\frac{29+\sqrt{489}}{12}\).
Reviewer: Olaf Ninnemann (Berlin)
Citations:
Zbl 0992.11057References:
| [1] | Beukers, F., The lattice-points of a \(n\)-dimensional tetrahedra, Indag. Math., 37, 5, 365-372 (1975) · Zbl 0312.10037 |
| [2] | Cappell, S. E.; Shaneson, J. L., Genera of algebraic varieties and counting lattice points, Bull. AMS (New Series), 30, 62-69 (1994) · Zbl 0847.14010 |
| [3] | Diaz, R.; Robins, S., The Ehrhart polynomial of a lattice polytope, Ann. Math., 145, 503-518 (1997) · Zbl 0874.52009 |
| [4] | Durfee, A. H., The signature of smoothings of complex surface singularities, Math. Ann., 232, 85-98 (1978) · Zbl 0346.32016 |
| [5] | Ehrhart, E., Sur un problème degé diophabtienne libéaire II, J. Reine Angew. Math., 227, 25-49 (1967) · Zbl 0155.37503 |
| [6] | G.H. Hardy, J.E. Littlewood, Some problems of diophantine approximation, in: Proceedings of the 5th International Congress of Mathematics, 1912, pp. 223-229.; G.H. Hardy, J.E. Littlewood, Some problems of diophantine approximation, in: Proceedings of the 5th International Congress of Mathematics, 1912, pp. 223-229. · JFM 44.0237.01 |
| [7] | Hardy, G. H.; Littlewood, J. E., The lattice points of a right-angled triangle, Proc. London Math. Soc., 20, 15-36 (1921) · JFM 48.0197.07 |
| [8] | Hardy, G. H.; Littlewood, J. E., The lattice points of a right-angled triangle (second memoir), Hamburg Math. Abh., 1, 212-249 (1922) |
| [9] | Kantor, J. M.; Khovanskii, A., Une application du Théoréme de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entier de \(R^d\), C.R. Acad. Sci. Paris I, 317, 501-507 (1993) · Zbl 0791.52012 |
| [10] | Lehmer, D. H., The lattice points of an \(n\)-dimensional tetrahedron, Duke Math. J., 7, 341-353 (1940) · Zbl 0024.14901 |
| [11] | Lin, K.-P.; Yau, S. S.-T., Analysis of sharp polynomial upper estimate of number of positive integral points in a 4-dimensional tetrahedra, J. Reine Angew. Math., 547, 207-234 (2002) · Zbl 0992.11057 |
| [12] | Lin, K.-P.; Yau, S. S.-T., A sharp upper estimate of the number of integral points in a 5-dimensional tetrahedron, J. Number Theory, 93, 191-205 (2002) · Zbl 1013.11064 |
| [13] | Lin, K.-P.; Yau, S. S.-T., Counting the number of integral points in general \(n\)-dimesional terahedra and bernoulli polynomials, Canad. Math. Bull., 46, 229-241 (2003) · Zbl 1056.11054 |
| [14] | M. Merle, B. Teissier, Conditions d’adjonction d’aprés Du Val, Sěminaire sur les singularites des surfaces, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980, pp. 229-245.; M. Merle, B. Teissier, Conditions d’adjonction d’aprés Du Val, Sěminaire sur les singularites des surfaces, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980, pp. 229-245. · Zbl 0461.14009 |
| [15] | Milnor, J.; Orlik, P., Isolated singularities defined by weighted homogeneous polynomials, Topology, 9, 385-393 (1970) · Zbl 0204.56503 |
| [16] | Mordell, L. J., Lattice points in tetrahedron and generalized Dedekind sums, J. Indian Math., 15, 41-46 (1951) · Zbl 0043.05101 |
| [17] | Morelli, R., Pick’s theorem and the Todd class of tori variety, Adv. in Math., 100, 183-231 (1993) · Zbl 0797.14018 |
| [18] | Pomerance, C., The Role of Smooth Numbers in Number Theoretic Algorithms (1994), ICM: ICM Zurich, pp. 411-422 · Zbl 0854.11047 |
| [19] | C. Pomerance, Lecture Notes on Primality Testing and Factoring: A Short Course, Kent State University, 1984.; C. Pomerance, Lecture Notes on Primality Testing and Factoring: A Short Course, Kent State University, 1984. · Zbl 0719.11002 |
| [20] | Pommershim, J., Toric variety, lattice points and Dedekind sums, Math. Ann., 295, 1-24 (1993) · Zbl 0789.14043 |
| [21] | Rosser, J. B., On the first case of Fermat’s last theorem, Bull. Amer. Math. Soc., 45, 636-640 (1939) · JFM 65.0143.03 |
| [22] | Spencer, D. C., On a Hardy-Littlewood problem of diophantine approximation, Math. Proc. Cambridge Philos. Soc., 35, 527-547 (1939) · Zbl 0022.30904 |
| [23] | Spencer, D. C., The lattice points of tetrahedra, J. Math. Phys., 21, 189-197 (1942) · Zbl 0060.11501 |
| [24] | Xu, Y.-J.; Yau, S. S.-T., A sharp estimate of the number of integral points in a tetrahedron, J. Reine Angew. Math., 423, 199-219 (1992) · Zbl 0734.11048 |
| [25] | Xu, Y.-J.; Yau, S. S.-T., Durfee conjecture and coordinate free characterization of homogeneous singularities, J. Differential Geom., 37, 375-396 (1993) · Zbl 0793.32016 |
| [26] | Xu, Y.-J.; Yau, S. S.-T., A sharp estimate of the number of integral points in a 4-dimensional tetrahedron, J. Reine Angew. Math., 473, 1-23 (1996) · Zbl 0844.11063 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
