Counting rational points of two classes of algebraic varieties over finite fields

Abstract

Let $ p $ stand for an odd prime and let $ \eta\in \mathbb Z^+ $ (the set of positive integers). Let $ \mathbb F_q $ denote the finite field having $ q = p^\eta $ elements and $ \mathbb F_q^* = \mathbb F_q\setminus \{0\} $. In this paper, when the determinants of exponent matrices are coprime to $ q-1 $, we use the Smith normal form of exponent matrices to derive exact formulas for the numbers of rational points on the affine varieties over $ \mathbb F_q $ defined by $ \left\{ \begin{aligned} &a_1x_1^{d_{11}}...x_n^{d_{1n}}+... +a_sx_1^{d_{s1}}...x_n^{d_{sn}} = b_1,\\ &a_{s+1}x_1^{d_{s+1,1}}...x_n^{d_{s+1,n}}+... +a_{s+t}x_1^{d_{s+t,1}}...x_n^{d_{s+t,n}} = b_2 \end{aligned} \right. $ and $ \left\{ \begin{aligned} &c_1x_1^{e_{11}}...x_m^{e_{1m}}+... +c_rx_1^{e_{r1}}...x_m^{e_{rm}} = l_1,\\ &c_{r+1}x_1^{e_{r+1,1}}...x_m^{e_{r+1,m}}+... +c_{r+k}x_1^{e_{r+k,1}}...x_m^{e_{r+k,m}} = l_2,\\ &c_{r+k+1}x_1^{e_{r+k+1,1}}...x_m^{e_{r+k+1,m}}+... +c_{r+k+w}x_1^{e_{r+k+w,1}}...x_m^{e_{r+k+w,m}} = l_3, \end{aligned} \right. $ respectively, where $ d_{ij}, e_{i'j'}\in \mathbb Z^+, a_i, c_{i'}\in \mathbb F_q^*, i = 1, ..., s+t,$ $j = 1, ..., n, i' = 1, ..., r+k+w, j' = 1, ..., m, $ and $ b_1, b_2, l_1, l_2, l_3\in \mathbb F_q $. These formulas extend the theorems obtained by Q. Sun in 1997. Our results also give a partial answer to an open question posed by S.N. Hu, S.F. Hong and W. Zhao [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015), 135–153].

Keywords

• finite field
• algebraic variety
• rational point
• prime number
• smith normal form