Counting rational points of quartic diagonal hypersurfaces over finite fields

Abstract

Let $ \mathbb{F}_q $ be the finite field of order $ q $ where $ q = p^{k} $, $ k $ is a positive integer and $ p $ is an odd prime. Let $ \mathbb{F}_q^* $ represent the nonzero elements of $ \mathbb{F}_{q} $. For $ f(x_1, \cdots, x_n)\in\mathbb{F}_q[x_1, \cdots, x_n] $, we used $ N\big(f(x_1, \cdots, x_n) = 0\big) $ to denote the number of $ \mathbb{F}_q $-rational points of the affine hypersurface $ f(x_1, \cdots, x_n) = 0 $. In 2020, Zhao et al. obtained the explicit formulae for $ N(x_1^4+x_2^4 = c) $, $ N(x_1^4+x_2^4+x_3^4 = c) $ and $ N(x_1^4+x_2^4+x_3^4+x_4^4 = c) $ over $ \mathbb{F}_q $, with $ c\in\mathbb{F}_q^* $. In this paper, by using Jacobi sums and an analog of the Hasse-Davenport theorem, we arrived at explicit formulae for $ N(a_1x_1^4+a_2x_2^4 = c) $ and $ N(b_1x_1^4+b_2x_2^4+b_3x_3^4 = c) $ with $ a_i, b_j\in \mathbb{F}_q^* (1\leq i \leq 2, 1 \leq j \leq 3) $ and $ c\in \mathbb{F}_q $. Furthermore, by using the reduction formula for Jacobi sums, the number of rational points of the quartic diagonal hypersurface $ a_1x_1^4+a_2x_2^4+\cdots+a_nx_n^4 = c $ of $ n\geq 4 $ variables with $ a_i\in\mathbb{F}_q^* $ $ (1\leq i\leq n) $, $ c\in\mathbb{F}_q $ and $ p\equiv1({\rm{mod}} \ 4) $, can also be deduced. These extended and improved earlier results.

Keywords

• finite field
• rational point
• diagonal equation
• jacobi sum