On the number of zeros of diagonal quartic forms over finite fields

Let F-q be the finite field of q = p(m) (math) 1 (mod 4) elements with p being an odd prime and m being a positive integer. For c, y is an element of F-q with y is an element of F-q* non-quartic, let N-n (c) and M-n (y) be the numbers of zeros of x(1)(4) + ... + x(n)(4) = c and x(1)(4 )+ ... + x(n-1)(4) + yx(n)(4) = 0, respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function Sigma(infinity)(n-1) N-n (0)x(n) is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions Sigma(infinity)(n=1) N-n(C)x(n) and Sigma(infinity)(n=1) Mn+1(y)x(n) are rational functions in x. We also obtain the explicit expressions of these generating functions. Our result extends Myerson's theorem gotten in 1979.