Results in Mathematics

, 74:37

# Counting Rational Points of an Algebraic Variety over Finite Fields

• Shuangnian Hu
• Xiaoer Qin
• Junyong Zhao
Article

## Abstract

Let $${\mathbb {F}}_q$$ denote the finite field of odd characteristic p with q elements ($$q=p^{n},n\in {\mathbb {N}}$$) and $${\mathbb {F}}_q^*$$ represent the nonzero elements of $${\mathbb {F}}_{q}$$. In this paper, by using the Smith normal form we give a formula for the number of rational points of the algebraic varieties defined by the following system of equations over $${\mathbb {F}}_{q}$$:
\begin{aligned} \left\{ \begin{array}{ll} \sum \limits _{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} \ldots x_{n_1}^{e^{(1)}_{i,n_1}} +\sum \limits _{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} \ldots x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ \sum \limits _{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} \ldots x_{n_3}^{e^{(2)}_{j,n_3}} +\sum \limits _{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} \ldots x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right. \end{aligned}
where the integers $$1\le r_1<r_2$$, $$1\le r_3<r_4$$, $$1\le n_1<n_2$$, $$1\le n_3<n_4$$, $$n_1\le n_3$$, $$b_1, b_2\in \mathbb {F}_{q}$$, $$a_{1i}\in \mathbb {F}_{q}^{*}(1\le i\le r_2)$$, $$a_{2j}\in \mathbb {F}_{q}^{*}(1\le j\le r_4)$$ and the exponent of each variable is positive integer. Our result provides a partial answer to an open problem raised in Hu et al. (J Number Theory 156:135–153, 2015).

## Keywords

Finite field algebraic variety rational point Smith normal form exponent matrix

11T06 11T71

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## Authors and Affiliations

• Shuangnian Hu
• 1
• Xiaoer Qin
• 2
• Junyong Zhao
• 3
1. 1.School of Mathematics and StatisticsNanyang Institute of TechnologyNanyangPeople’s Republic of China
2. 2.College of Mathematics and StatisticsYangtze Normal UniversityChongqingPeople’s Republic of China
3. 3.Mathematical CollegeSichuan UniversityChengduPeople’s Republic of China