The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

• 来源期刊: 代数集刊（英文版）
• 关键词: finite field variety rational point exponent matrix Smith normal form
• 年，卷（期）: 2017,（4）
• 页码范围: 705-720
• 页数: 16页
• 语种: 英文
• DOI: 10.1142/S1005386717000475

摘要

Let Fq stand for the finite field of odd characteristic p with q elements (q =pn,n ∈ N) and F*q denote the set of all the nonzero elements of Fq.Let m and t be positive integers.By using the Smith normal form of the exponent matrix,we obtain a formula for the number of rational points on the variety defined by the following system of equations over Fq:t-1∑j=0 rj+1-rj∑i=1 ak,rj+ix1e(k)rj+i,1…xe(k)rj+i,nj+1nj+1=bk,k=1,…m,where the integers t ＞ 0,r0 =0 ＜ r1 ＜ r2 ＜ … ＜ rt,1 ≤ n1 ＜ n2 ＜ … ＜ nt and 0 ≤ j ≤ t-1,bk ∈ Fq,ak,i ∈ F*q (k =1,...,m,i =1,...,rt),and the exponent of each variable is a positive integer.Further,under some natural conditions,we arrive at an explicit formula for the number of Fq-rational points on the above variety.It extends the results obtained previously by Wolfmann,Sun,Wang,Hong et al.Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields,J.Number Theory 156 (2015) 135-153].