The number of lattice points in an 𝑛-dimensional tetrahedron

Abstract

is a polynomial 7with rational coefficients in the quantities v1/A and the as. One should mention here that both theorems are completely trivial for n = 1. Theorem 2 is also trivial for n =2, and indeed for the following reason. Clearly if _ 0 (mod aja2), aix, + a2X2= implies that xi 0 (mod a2), x2 0 (mod a,). Let xi = a2x1', x2= ax2'. Hence we see that if _= 0 (mod a1a2) the number of solutions of (3) alxl + a2x2 = ?, xi _ 0, x2 > O is the same as the number of solutions of