Factorization formulae on counting zeros of diagonal equations over finite fields

Let $N$ be the number of solutions $(u_1,\ldots,u_n)$ of the equation $a_1u_1^{d_1}+\cdots+a_nu_n^{d_n}=0$ over the finite field $F_q$, and let $I$ be the number of solutions of the equation $\sum_{i=1}^nx_i/d_i\equiv 0\pmod{1}, 1\leqslant x_i\leqslant d_i-1$. If $I>0$, let $L$ be the least integer represented by $\sum_{i=1}^nx_i/d_i, 1\leqslant x_i\leqslant d_i-1$. $I$ and $L$ play important roles in estimating $N$. Based on a partition of $\{d_1,\dots,d_n\}$, 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.