Mathematics

CLC number: O156

Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields

¹ School of Mathematics and Physics, Nanyang Institute of Technology, Nanyang 473004, Henan, China
² Faculty of Science and Technology, Beijing Normal University-Hong Kong Baptist University United International College, Zhuhai 519087, Guangdong, China
³ School of Information Engineering, Nanyang Institute of Technology, Nanyang 473004, Henan, China

Received: 5 June 2023

Abstract

Let ${\mathbf{F}}_q$ be the finite field, $q=p^k$, with $p$ being a prime and $k$ being a positive integer. Let ${\mathbf{F}}_q^{\mathrm{*}}$ be the multiplicative group of ${\mathbf{F}}_q$, that is ${\mathbf{F}}_q^{\mathrm{*}}={\mathbf{F}}_q\backslash \{\mathrm{0}\}$. In this paper, by using the Jacobi sums and an analog of Hasse-Davenport theorem, an explicit formula for the number of solutions of cubic diagonal equation $x_{\mathrm{1}}^{\mathrm{3}}+x_{\mathrm{2}}^{\mathrm{3}}+\cdots +x_n^{\mathrm{3}}=c$ over ${\mathbf{F}}_q$ is given, where $c\in {\mathbf{F}}_q^{\mathrm{*}}$ and $p\equiv \mathrm{1}(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{3})$. This extends earlier results.