Factorization formulae on counting zeros of diagonal equations over finite fields

Let N N be the number of solutions ( u 1 , … , u n ) (u_1,\ldots ,u_n) of the equation a 1 u 1 d 1 + ⋯ + a n u n d n = 0 a_1u_1^{d_1}+\cdots +a_nu_n^{d_n}=0 over the finite field F q F_q , and let I I be the number of solutions of the equation ∑ i = 1 n x i / d i ≡ 0 ( mod 1 ) , 1 ⩽ x i ⩽ d i − 1 \sum _{i=1}^nx_i/d_i\equiv 0\pmod {1}, 1\leqslant x_i\leqslant d_i-1 . If I > 0 I>0 , let L L be the least integer represented by ∑ i = 1 n x i / d i , 1 ⩽ x i ⩽ d i − 1 \sum _{i=1}^nx_i/d_i, 1\leqslant x_i\leqslant d_i-1 . I I and L L play important roles in estimating N N . Based on a partition of { d 1 , … , d n } \{d_1,\dots ,d_n\} , we obtain the factorizations of I , L I, L and N N , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for N N in some special cases.