An upper estimate of integral points in real simplices with an application to singularity theory

Let ∆(a1, a2, · · · , an) be an n-dimensional real simplex with vertices at (a1, 0, · · · , 0), (0, a2, · · · , 0), · · · , (0, 0, · · · , an). Let P(a1,a2,··· ,an) be the number of positive integral points lying in ∆(a1, a2, · · · , an). In this paper we prove that n!P(a1,a2,··· ,an) ≤ (a1 − 1)(a2 − 1) · · · (an − 1). As a consequence we have proved the Durfee conjecture for isolated weighted homogeneous singularities: n!pg ≤ μ, where pg and μ are the geometric genus and Milnor number of the singularity, respectively.

[1]  Alan Durfee.  The signature of smoothings of complex surface singularities , 1978 .

[2]  S. Yau,et al.  Counting the Number of Integral Points in General $n$ -Dimensional Tetrahedra and Bernoulli Polynomials , 2003, Canadian Mathematical Bulletin.

[3]  Sinai Robins,et al.  The Ehrhart polynomial of a lattice polytope , 1997 .

[4]  A. Barvinok,et al.  An Algorithmic Theory of Lattice Points in Polyhedra , 1999 .

[5]  D. Spencer.  The Lattice Points of Tetrahedra , 1942 .

[6]  Shaofang Hong,et al.  Analysis of sharp polynomial upper estimate of number of positive integral points in a five-dimensional tetrahedra , 2008, Discret. Math..

[7]  L. J. Mordell,et al.  Lattice Points in a Tetrahedron and Generalized Dedekind Sums , 1951 .

[8]  J. E. Littlewood,et al.  Some problems of diophantine approximation , 1914 .

[9]  Michèle Vergne,et al.  Lattice points in simple polytopes , 1997 .

[10]  A. I. Barvinok,et al.  Computing the Ehrhart polynomial of a convex lattice polytope , 1994, Discret. Comput. Geom..

[11]  J. Littlewood,et al.  On a Hardy-Littlewood problem of diophantine approximation , 1939, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  Carl Pomerance,et al.  The Role of Smooth Numbers in Number Theoretic Algorithms , 1995 .

[13]  吉永 悦男.  Isolated singularities defined by weighted homogeneous polynomials , 1984 .

[14]  Kyoji Saito,et al.  Quasihomogene isolierte Singularitäten von Hyperflächen , 1971 .

[15]  B. Teissier,et al.  Conditions d'adjonction, d'après du val , 1980 .

[16]  G. H. Hardy,et al.  Some problems of Diophantine approximation: The lattice-points of a right-angled triangle. (Second memoir.) , 1922 .

[17]  Sylvain E. Cappell,et al.  Genera of algebraic varieties and counting of lattice points , 1994, math/9401219.

[18]  S. Yau,et al.  A sharp estimate of the number of integral points in a tetrahedron. , 1992 .

[19]  James Pommersheim,et al.  Toric varieties, lattice points and Dedekind sums , 1993 .