On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

Journal of Number Theory - Tập 122 - Trang 184-210 - 2007
Xuejun Wang1, Stephen Yau1
1Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60206, USA

Tài liệu tham khảo

Barnes, 1904, On the theory of the multiple Gamma function, Trans. Cambridge Philos. Soc., XIX, 374

Barvinok, 1994, Computing the Ehrhart polynomial of a convex lattice polytope, Discrete Comput. Geom., 12, 35, 10.1007/BF02574364

Barvinok, 1999, An algorithmic theory of lattice points in polyhedra, vol. 38

Beck, 1999, A closer look at lattice points in rational simplices, Electron. J. Combin., 6, 10.37236/1469

Beck, 2000, Counting lattice points by means of the residue theorem, Ramanujan J., 4, 299, 10.1023/A:1009853104418

Beck, 2002, The Frobenius problem, rational polytopes, and Fourier–Dedekind sums, J. Number Theory, 96, 1

Brion, 1997, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Amer. Math. Soc., 10, 797, 10.1090/S0894-0347-97-00242-7

Cappell, 1994, Genera of algebraic varieties and counting of lattice points, Bull. Amer. Math. Soc. (N.S.), 30, 62, 10.1090/S0273-0979-1994-00436-7

Comtet, 1974

Danilov, 1978, The geometry of toric varieties, Russian Math. Surveys, 33, 97, 10.1070/RM1978v033n02ABEH002305

Diaz, 1997, The Ehrhart polynomial of a lattice polytope, Ann. of Math., 145, 503, 10.2307/2951842

Durfee, 1978, The signature of smoothings of complex surface singularities, Math. Ann., 232, 85, 10.1007/BF01420624

Ehrhart, 1967, Sur un probleme de geometrie diophantienne lineaire II, J. Reine Angew. Math., 227, 25

Fulton, 1984

Granville, 1991, The lattice points of an n-dimensional tetrahedron, Aequationes Math., 41, 234, 10.1007/BF02227458

G.H. Hardy, J.E. Littlewood, Some problems of Diophantine approximation, in: Proc. 5th Int. Congress of Mathematics, 1912, pp. 223–229

Hardy, 1921, Some problems of Diophantine approximation: The lattice points of a right-angled triangle, Proc. London Math. Soc. (2), 20, 15

Hardy, 1922, Some problems of Diophantine approximation: The lattice points of a right-angled triangle (second memoir), Hamburg Math. Abh., 1, 212

Kantor, 1993, Une application du Theorem de Riemann–Roch combinatoire au polynome d'Ehrhart des polytopes entier de Rn, C. R. Acad. Sci. Paris Ser. I, 317, 501

Lehmer, 1940, The lattice points of an n-dimensional tetrahedron, Duke Math. J., 7, 341, 10.1215/S0012-7094-40-00719-0

Ke-Pao Lin, S.S.-T. Yau, Classification of affine varieties being cones over nonsingular projective varieties: Hypersurface case, preprint, 1999

Lin, 2002, A sharp estimate of number of integral points in a 5-dimensional tetrahedra, J. Number Theory, 93, 207, 10.1006/jnth.2001.2720

Lin, 2002, Analysis of sharp polynomial upper estimate of number of positive integral points in a 4-dimensional tetrahedra, J. Reine Angew. Math., 547, 191

Lin, 2003, Counting number of integral points in a general n-dimensional tetrahedra and Bernoulli polynomials, Canad. Math. Bull., 24, 229, 10.4153/CMB-2003-023-0

McMullen, 1978, Lattice invariant valuations on rational polytopes, Arch. Math. (Basel), 31, 509, 10.1007/BF01226481

McMullen, 1983, Valuations on convex bodies, 170

Merle, 1980, Conditions d'adjonction d'après Du Val, vol. 777, 229

Milnor, 1970, Isolated singularities defined by weighted homogeneous polynomials, Topology, 9, 385, 10.1016/0040-9383(70)90061-3

Mordell, 1951, Lattice points in a tetrahedron and generalized Dedekind sums, J. Indian Math. Soc., 15, 41

Morelli, 1993, Picks theorem and the Todd class of a toric variety, Adv. Math., 100, 183, 10.1006/aima.1993.1033

Pick, 1899, Geometrisches zur Zahlentheorie, Sitzenber. Lotos (Prague), 19, 311

Pommersheim, 1993, Toric variety, lattice points and Dedekind sums, Math. Ann., 295, 1, 10.1007/BF01444874

Rademacher, 1973

Rosser, 1939, On the first case of Fermat's last theorem, Bull. Amer. Math. Soc., 45, 636, 10.1090/S0002-9904-1939-07058-4

Spencer, 1939, On a Hardy–Littlewood problem of Diophantine approximation, Math. Proc. Cambridge Philos. Soc., 35, 527, 10.1017/S0305004100021320

Spencer, 1942, The lattice points of tetrahedra, J. Math. Phys., 21, 189, 10.1002/sapm1942211189

Xu, 1992, A sharp estimate of number of integral points in a tetrahedron, J. Reine Angew. Math., 423, 199

Xu, 1993, Durfee conjecture and coordinate free characterization of homogeneous singularities, J. Differential Geom., 37, 375, 10.4310/jdg/1214453681

Xu, 1996, A sharp estimate of number of integral points in a 4-dimensional tetrahedra, J. Reine Angew. Math., 473, 1

S.S.-T. Yau, L. Zhang, An upper estimate on integral points in real simplices with an application in singularity theory, Math. Res. Lett., in press

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Journal of Number Theory

Tập 122

184-210

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