有限域上一类方程组解数的直接公式

An Explicit Formula for the Number of Solutions of Certain System of Equations over Finite Fields

本文给出有限域F=F_q(q=p^f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1^(s_r)∑_(j=1)^(m_i-m_(i-1))a_(m_(i-1)+j)x_1^(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F^(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．

Let F=Fq（q=p^f,f≥1,p is an odd prime number）.In this paper,we get an explicit formula for the number of F-rational solutions of the following systems of equations defined over ∑_（i=s_（r-1）＋1^sr∑_（j=1）^mi-mi-1ami-1＋jx1^dmi-1＋j,1…xni^dmi-1＋j,ni=br,r=1,…,k where dij〉0 satisfying certain conditions,ai∈F^＊,bi∈F,0= s0〈s1〈…〈sk,0=m0〈m1〈…〈msk,0=n0o〈n1〈…〈nsk, m1≤n1,…,msk≤nsk.