Number of Zeros of Diagonal Polynomials over Finite Fields,

Let F be a finite field with q=pf elements, where p is a prime. Let N be the number of solutions (x1,…,xn) of the equation c1xd11+•••+cnxdnn=c over the finite fields, where d1 q−1, ciεF*(i=1, 2,…,n), and cεF. In this paper, we prove that if b1 is the least integer such that b1≥∑ni=1 (f/ri) (Di, p−1)/(p−1), then q[b1/f]−1 N, where ri is the least integer such that di pri−1, Didi=pri−1, the (Di, p−1) denotes the greatest common divisor of Di and p−1, [b1/f] denotes the integer part of b1/f. If di=d, then this result is an improvement of the theorem that pb N, where b is an integer less than n/d, obtained by J. Ax (1969, Amer. J. Math.86, 255–261) and D. Wan (1988, Proc. AMS103, 1049–1052), under a certain natural restriction on d and n.