ON THE NUMBER OF SOLUTIONS TO CERTAIN DIAGONAL EQUATIONS OVER FINITE FIELDS

We consider a diagonal equation, which can be reduced to the form x(1)(2) + center dot center dot center dot + x(n-2)(2) + x(n-1)(k) + x(n)(k) = 0, over a finite field of characteristic p > 2. In 1997, Sun obtained the explicit formula for the number of solutions to an equation of this type when n is even. In this paper, we find explicit formulas for the number of solutions when n is odd, k = 2(r) h, and there exists a positive integer l such that p(l) 2(m-1) h + 1(mod 2(m) h), m = 3 or 4, r >= m, h = 1 or 3.