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

Let a⩾b⩾c⩾d⩾e⩾1a⩾b⩾c⩾d⩾e⩾1 be real numbers and P5P5 be the number of positive integral solutions of xa+yb+zc+ud+ve⩽1. In this paper we show that 120P5⩽(a-1)(b-1)(c-1)(d-1)(e-1).120P5⩽(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 P5P5 given by Lin and Yau is strictly sharper than that suggested by Durfee conjecture if e⩾29+48912, but is not sharper than that suggested by Durfee conjecture if 4⩽e<29+48912.