关于有限域上一类方程解的个数

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N(a1x12 + ···+anxn2 = bx1 ···xs) denote the number of solutions(x1,...,xn) of the equation a1x12 +···+ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F*q,b ∈ F*q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 + ··· + anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.

On the Number of Solutions of Certain Equations over Finite Fields

Let $\F_q$ be a finite field with $q=p^f$ elements, where $p$ is an odd prime. Let $N(a_1x_1^2+\cdots+a_nx_n^2=bx_1\cdots x_s)$ denote the number of solutions $(x_1,\ldots,x_n)$ of the equation $a_1x_1^2+\cdots +a_nx_n^2=bx_1\cdots x_s$ in $\F_q^n$, where $n>5$, $s5$, $3\leq s