ON THE DIOPHANUNE EQUATION $\[\sum\limits_{i = 0}^N {\frac{{{x_i}}}{{{d_i}}}} \equiv 0\]$ (mod 1) AND ITS APPLICATIONS

The number $\A({d_1}, \cdots ,{d_n})\]$ of solutions of the equation $$\\sum\limits_{i = 0}^n {\frac{{{x_i}}}{{{d_i}}}} \equiv 0(\bmod 1),0 < {x_i} < {d_i}(i = 1,2, \cdots ,n)\]$$ where all the $\{d_i}s\]$ are positive integers, is of significance in the estimation of the number $\N({d_1}, \cdots {d_n})\]$ of solutiohs in a finite field $\{F_q}\]$ of the equation $$\\sum\limits_{i = 1}^n {{a_i}x_i^{{d_i}}} = 0,{x_i} \in {F_q}(i = 1,2, \cdots ,n)\]$$ where all the $\a_i^''s\]$ belong to $\F_q^*\]$. the multiplication group of $\F_q^{1,2]}\]$. In this paper, applying the inclusion-exclusion principle, a greneral formula to compute $\A({d_1}, \cdots ,{d_n})\]$ is obtained. For some special cases more convenient formulas for $\A({d_1}, \cdots ,{d_n})\]$ are also given, for example, if $\{d_i}|{d_{i + 1}},i = 1, \cdots ,n - 1\]$, then $$\A({d_1}, \cdots ,{d_n}) = ({d_{n - 1}} - 1) \cdots ({d_1} - 1) - ({d_{n - 2}} - 1) \cdots ({d_1} - 1) + \cdots + {( - 1)^n}({d_2} - 1)({d_1} - 1) + {( - 1)^n}({d_1} - 1).\]$$