有限域上一类方程组的解数公式

The number of solutions to a class of equation systems over finite fields

设F_q为一个q元有限域,其中q=p^s(s≥1),p是一个奇素数.本文给出下列方程组在F_q上的解数公式:a_(k1)x_1^(d_(11)^((k)))...x_(n_1)^(d_(1n_1)^((k)))+...+a_(k,s_1)x_1^(d_(s_1,1)^((k)))...x_(n_1)^(d_(s_1,n_1)^((k)))+a_(k,s_1)+1x_1^(d_(s_1+1,1)^((k)))...x_(n_2)^(d_(s_1+1,n_2)^((k)))+...a_(k,s_2)x_1^(d_(s_2,1)^((k)))...x_(n_2)^(d_(s_2,1)^((k)))...x_(n_2)^(d_(s_2,n_2)^((k)))=b_k,k=1,...,m,其中0<s_1<s_2,0<n_1<n_2,a_(ki)∈F_q~*,b_k∈F_q,d_(ij)^(k)>0(k=l,...,m,i=1,...,s_2,j=1,...,n_2).特别当ms_1≤n_1,ms_2≤n_2,d_(ij)^(k)满足一定条件时,得到了明确的解数公式.

Let Fq be a finite field with q = ps （s ≥ 1） elements, where p is an odd prime number. In this paper, we present a formula for the number of solutions to the following equation system defined over Fq： where 0 〈 sx 〈 s2, 0 〈 n1 〈 n2, aki∈ Fq, bk ∈ Fq, dij（k） 〉 0 （k = 1,...,m, i = 1,...,s2, j = 1,...,n2）. Especially when ms1 ≤ nx, ms2 ≤ n2, dij（k） satisfying certain conditions, an explicit formula is obtained for the number of solutions to the equation system.