Analysis of sharp polynomial upper estimate of number of positive integral points in a five-dimensional tetrahedra

Let a ≥ b ≥ c ≥ d ≥ e ≥ 1 be real numbers and P5 be the number of positive integral solutions of frac(x, a) + frac(y, b) + frac(z, c) + frac(u, d) + frac(v, e) ≤ 1. In this paper we show that 120 P5 ≤ (a - 1) (b - 1) (c - 1) (d - 1) (e - 1) . This confirms a conjecture of Durfee for the dimension 5 case. We show also that the upper estimate of P5 given by Lin and Yau is strictly sharper than that suggested by Durfee conjecture if e ≥ frac(29 + sqrt(489), 12), but is not sharper than that suggested by Durfee conjecture if 4 ≤ e < frac(29 + sqrt(489), 12). © 2007 Elsevier B.V. All rights reserved.