Factorization formulae on counting zeros of diagonal equations over finite fields

Let N be the number of solutions (u(1), . . . , u(n)) of the equation a(1)u(1)(d1) + (.) (.) (.) + a(n)u(n)(dn) = 0 over the. nite. eld F-q, and let I be the number of solutions of the equation Sigma(n)(i=1) x(i)/d(i) equivalent to 0 (mod 1), 1 <= x(i) <= d(i) - 1. If I > 0, let L be the least integer represented by Sigma(n)(i=1) x(i)/d(i), 1 <= x(i) <= d(i) - 1. I and L play important roles in estimating N. Based on a partition of {d(1), . . . , d(n)}, we obtain the factorizations of I, L and N, respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for N in some special cases.