The Number of Solutions of Certain Equations over Finite Fields

• 作者: HU Shuangnian LIU Jianghan QIN Zhentao
• 来源期刊: 武汉大学自然科学学报（英文版）
• 学科: 数学
• 年，卷（期）: 2022,（1）
• 所属期刊栏目: Mathematics
• 页码范围: 49-52
• 页数: 4页
• 分类号: O156
• 语种: 英文
• DOI: 10.1051/wujns/2022271049

摘要

Let sbea positive integer,p be an odd prime,q=ps,and let Fq be a finite field of q elements.Let Nq be the number of solutions of the following equations:(xm11+xm22+…+xmnn)k=x1x2…xnx(kn+1)(n+1)…xk1t over the finite field Fq,with n≥2,t＞n,k,and kj(n+1≤j≤t),mi(l≤i≤n)are positive integers.In this paper,we find formulas for Nq when there is a positive integer l such that dD|(pl+1),where D=lcm[d1,…,dn],d=gcd(n∑(n=1)M/mi-kM,(q-1)/D),M=lcm[m1,…,mn],dj=gcd(mj,q-1),1≤j≤n.And we determine Nq explicitly under certain cases.This extends Markoff-Hurwitz-type equations over finite field.