IJPAM: Volume 50, No. 4 (2009)

ON THE NUMBER OF SOLUTIONS OF THE EQUATION
$a_1x_1^2+\cdots +a_nx_n^2=bx_1\cdots x_s$ IN FINITE FIELDS

Luo Yanmei$^1$, Zhao Zhengjun$^2$, Cao Xiwang$^3$
$^{1,2,3}$Department of Mathematics
Nanjing University of Aeronautics and Astronautics
Nanjing, 210016, P.R. CHINA
$^1$e-mail: ymluo@nuaa.edu.cn
$^2$e-mail: zzj0608115@126.com
$^3$e-mail: xwcao@nuaa.edu.cn


Abstract.Let $\mathbb{F}_q$ be a finite field with $q=p^f$ elements, where $p$ is an odd prime. Let $N(a_1x_1^2+\cdots +a_nx_n^2=bx_1\cdots x_s)$ denote the number of solutions of the equation

\begin{eqnarray*} a_1x_1^2+\cdots +a_nx_n^2=bx_1\cdots x_s \end{eqnarray*}

in the finite field $\mathbb{F}_q$. We obtain the explicit formula for $N(a_1x_1^2+\cdots +a_nx_n^2=bx_1\cdots x_s)$, where $n\geq3$, $s>n$, $a_i\in{\mathbb{F}_q}^* $, $b\in {\mathbb{F}_q}^*$. We also find explicit formulas for the number of solutions of $a_1x_1^2+\cdots +a_nx_n^2=bx_1\cdots x_s$, where $s< 3$ or $n\leq 5$ and $s\leq 4$.

Received: November 11, 2008

AMS Subject Classification: 11T06, 11T23

Key Words and Phrases: finite fields, solutions of equation, quadratic character

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 50
Issue: 4