The number of solutions of cubic diagonal equations over finite fields

Let p be a prime, k be a positive integer, q = p(k), and F-q be the finite field with q elements. Let F-q(*) be the multiplicative group of Fq, that is F-q(*) = F-q / {0} . In this paper, explicit formulae for the numbers of solutions of cubic diagonal equations alpha(1)x(1)(3) + alpha(2)x(2)(3) = c and b(1)x(1)(3) + b(2)x(2)(3) + b(3)x(3)(3) = c over F-q are given, with alpha(i); b(j) 2 F*(q) (1 <= i <= 2; 1 <= j <= 3), c is an element of 2 F-q and p equivalent to 1(mod 3). Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the cubic diagonal equations alpha(1)x(1)(3) + alpha(2)x(2)(3) + (. . .) + alpha(s)x(s)(3) = c of s >= 4 variables with alpha(i) is an element of 2 F*(q) (1 <= i <= s), is an element of 2 F-q and p equivalent to 1(mod 3), can also be deduced.