The number of solutions of cubic diagonal equations over finite fields

Abstract

<abstract><p>Let $ p $ be a prime, $ k $ be a positive integer, $ q = p^k $, and $ \mathbb{F}_q $ be the finite field with $ q $ elements. Let $ \mathbb{F}_q^* $ be the multiplicative group of $ \mathbb{F}_{q} $, that is $ \mathbb{F}_q^* = \mathbb{F}_{q}\setminus\{0\} $. In this paper, explicit formulae for the numbers of solutions of cubic diagonal equations $ a_1x_1^3+a_2x_2^3 = c $ and $ b_1x_1^3+b_2x_2^3+b_3x_3^3 = c $ over $ \mathbb{F}_q $ are given, with $ a_i, b_j\in\mathbb{F}_q^* $ $ (1\leq i\leq 2, 1\leq j\leq 3) $, $ c\in\mathbb{F}_q $ and $ p\equiv1(\rm{mod} \ 3) $. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the cubic diagonal equations $ a_1x_1^3+a_2x_2^3+\cdots+a_sx_s^3 = c $ of $ s\geq 4 $ variables with $ a_i\in\mathbb{F}_q^* $ $ (1\leq i\leq s) $, $ c\in\mathbb{F}_q $ and $ p\equiv1(\rm{mod} \ 3) $, can also be deduced.</p></abstract>