The number of rational points of certain quartic diagonal hypersurfaces over finite fields

Let p he an odd prime and let F-q be a finite field of characteristic p with order q = p(s). For f((X1),..., X-n) is an element of F-q[X-1,..., X-n] we denote by N(f(X-1,...,X-n) = 0) the number of F-q -rational points on the affine hypersurface f(X-1,...,X-n) = 0. In 1981, Myerson gave a formula for N(X-1(4 )+ ... + X-n(4)= 0). Recently, Zhao and Zhao obtained an explicit formula for N(X-1(4) + X-2(4) = c) with c is an element of F-q* :=F-q \ {0}. In this paper, by using the Gauss sum and Jacobi sum, we arrive at explicit formulas for N(X-1(4) + X-2(4) + X-3(4) = c) and N(X-1(4) + X-2(4) + X-3(4) +X-4(4) = c) with c is an element of F-q*.