设F_q为有限域,f_i(x)=a_(i1)x_1~(d_(i1))+…+a_(in)x_n~(d_(in))+c_i(i=1,…,m)为F_q上一组对角多项式,用N(V)表示由f_i(i=1,…,m)确定的簇中的F_q．有理点的个数．通过应用Adolphson和Sperber所引进的牛顿多面体方法,证明了ord_qN(V)≥[1/d_1+…+1/d_n]-m,其中d_i=max{d_(1i),…,d_(mi)}．该结果在许多情形下可以改进Ax- Katz定理,并推广了Wan在m=1时得到的一个定理,而且我们对Wan的定理给出了一个不同的证明．

Let F_q be the finite field,and let N(V) denote the number of F_q-rational points on the variety defined by the diagonal polynomials over F_q:f_i(x)=a_(i1)x_1~(d_(i1))+…+a_(in)x_n~(d_(in))+C_i,i=1,...,m.By using the Newton polyhedra technique introduced by Adolphson and Sperber,we show that ord_qN(V)≥「1/(d_1)+…+1/(d_n)(?)-m with d_i=max{d_(1i),...,d_(mi)},which can improve the Ax-Katz theorem in many cases.This generalizes Wan's theorem for the case m=1.Moreover,we provide a different proof to Wan's theorem.