Mathematics

CLC number: O156

The Number of Solutions of Certain Equations over Finite Fields

¹ School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, Henan, China
² College of Computer and Information Science, Southwest University, Chongqing 400715, China
³ School of Mathmatics and Computer, Panzhihua University, Panzhihua 617000, Sichuan, China

Received: 2 November 2021

Abstract

Let s be a positive integer, p be an odd prime, $q=p^s$, and let $F_q$ be a finite field of q elements. Let $N_q$ be the number of solutions of the following equations: ${(x_1^{m_1}+x_2^{m_2}+\cdots +x_n^{m_n})}^k=$$x_1{x}_2\cdots x_n{x}_{n+1}^{k_{n+1}}\cdots x_t^{k_t}$ over the finite field $F_q$, with $n\ge 2,$$t>$$n,k,$ and $k_j(n+1\le j$$\le t),m_i(1\le i\le n)$ are positive integers. In this paper, we find formulas for $N_q$ when there is a positive integer l such that dD|($p^l+1$) , where $D=\text{lcm}$$\text{[}{d}_1,\cdots ,d_n\text{]},$$d\text{\hspace{0.17em}}=\text{\hspace{0.17em}gcd}({\displaystyle \sum _{i=1}^{n}M/m_i}-kM,(q-1)/D),M=\text{lcm}$$\text{[}{m}_1,\cdots ,m_n\text{]},$$d_j=$$\text{gcd}$$\text{(}{m}_j\text{,}q-1\text{),}$$1\le j\le n.$ And we determine $N_q$ explicitly under certain cases. This extends Markoff-Hurwitz-type equations over finite field.