The number of rational points of some classes of algebraic varieties over finite fields

1 Introduction

Let Z + denote the set of positive integers. Let F q be the finite field of characteristic p , and let F q * ≔ F q \ { 0 } be its multiplicative group. We denote ∣ S ∣ by the number of elements of the finite set S . For any m ∈ Z + , we define ⟨ m ⟩ ≔ { 1 , 2 , … , m } . Let f i ( x 1 , … , x n ) ( i ∈ ⟨ m ⟩ ) be a polynomial with n variables over F q and let V ( f 1 , … , f m ) denote the algebraic variety defined by the simultaneous vanishing of f i ( x 1 , … , x n ) ( i ∈ ⟨ m ⟩ ) . Let N ( V ) stand for the number of F q -rational points on the algebraic variety V ( f 1 , … , f m ) in F q n . That is,

Particularly, we denote N ( f ) for N ( V ) if m = 1 . Determining the explicit value of N ( V ) is an important subject in finite fields. In general, it is difficult to present an explicit formula for N ( V ) . It is well known that there is an exact formula for the number N ( f ) when deg ( f ) ≤ 2 (see, for example, pp. 275–289 of [1]). Finding the formula for N ( V ) and relevant topics has attracted a lot of scholars in recent decades [2–24].

Sun [18] studied the number of rational points ( x 1 , … , x n ) ∈ F q n on the following affine hypersurface

with e i j ∈ Z + , a i ∈ F q * , b ∈ F q , 1 ≤ i , j ≤ n by proving that if gcd ( det ( e i j ) , q − 1 ) = 1 , then

where for any positive integers s , we have

and

Zhu et al. [24] considered a special variety defined by two or three equations, which is taken from the one investigated by Sun [18]. Sun’s result [18] was extended by Wang and Sun [19] by presenting a formula for the number of rational points ( x 1 , … , x n 2 ) ∈ F q n 2 on the affine hypersurface

with d i j ∈ Z + , a i ∈ F q * , 1 ≤ i , j ≤ n 2 . In 2015, Hu et al. [15] gave an uniform generalization to the results of [18] and [19]. In fact, they used the Smith normal form to present an explicit formula for N ( f ) of rational points ( x 1 , … , x n t ) ∈ F q n t on the hypersurface defined by

where the integers t > 0 , 0 = r 0 < r 1 < r 2 < … < r t , 1 ≤ n 1 < n 2 < … < n t , b ∈ F q , a i ∈ F q * , and e i j ∈ Z + , i ∈ ⟨ r t ⟩ , j ∈ ⟨ n t ⟩ . Zhu and Hong [23] followed the approach of [15] and gave an exact formula for the number of rational points on certain algebraic variety V = V ( f 1 , f 2 ) over F q as follows:

where b i ∈ F q , i = 1 , 2, t ∈ Z + , 0 = n 0 < n 1 < n 2 < … < n t , n k − 1 < n ≤ n k for some 1 ≤ k ≤ t , 0 = r 0 < r 1 < r 2 < … < r t , a i ( 1 ) , a i ′ ( 2 ) ∈ F q * , i ∈ ⟨ r ⟩ , i ′ ∈ ⟨ r t ⟩ , and the exponent of each variable is a positive integer.

Motivated by the works of [15,18,19,23], we consider the questions of counting F q -rational points of the variety V ( f 1 , f 2 ) determined by

and the variety V ( f 1 , f 2 , f 3 ) determined by

where e i j , d i ′ j ′ ∈ Z + , a i , c i ′ ∈ F q * , 1 ≤ i , j ≤ m 2 , 1 ≤ i ′ , j ′ ≤ n 3 , and b 1 , b 2 , l 1 , l 2 , l 3 ∈ F q .

Let

with e i j ( 1 ≤ i , j ≤ m 2 ) being given as in (1.3), and let

with d i ′ j ′ ( 1 ≤ i ′ , j ′ ≤ n 3 ) being given as in (1.4).

The main results of this article can be stated as follows.

This article is organized as follows. We present in Section 2 two preliminary lemmas that are needed in the proofs of our main results. Subsequently, we give the proof of Theorem 1.1 in Section 3. Section 4 is devoted to the proof of Theorem 1.2. In Section 5, we provide two examples to illustrate the validity of Theorems 1.1 and 1.2.

2 Auxiliary lemmas

In this section, we present two lemmas, which are needed in the proofs of our main results. We begin with a result due to Sun [18].

We also need the following result due to Zhu and Hong [23].

3 Proof of Theorem 1.1

In this section, we present the proof of Theorem 1.1. Firstly, we introduce some definitions and notations, which will be used in proving Theorem 1.1.

Let

with e i j ( 1 ≤ i , j ≤ m 1 ) being given as in (1.3). It is well known [25] that there are unimodular matrices U 1 , V 1 , U 2 , and V 2 such that

and

where E is given as in (1.5)

and

with v and v ′ being the ranks of the matrices E 1 and E , respectively. All elements

satisfy that g i ( E 1 ) ∣ g i + 1 ( E 1 ) ( i ∈ ⟨ v − 1 ⟩ ) and g i ( E ) ∣ g i + 1 ( E ) ( i ∈ ⟨ v ′ − 1 ⟩ ) . We say that the diagonal matrices D 1 0 0 0 in (3.1) and D 2 0 0 0 in (3.2) are the Smith normal form of the matrices E 1 and E and are abbreviated as SNF( E 1 ) and SNF( E ), respectively.

Set α ∈ F q * to be a primitive element of F q , for any β ∈ F q * , there is a unique integer γ ∈ [ 1 , q − 1 ] such that β = α γ , where γ is called the index of β with regard to the primitive element α and denoted by ind α β ≔ γ .

Let M 1 denote the number of rational points ( u 1 , … , u m 1 ) ∈ ( F q * ) m 1 on the affine hypersurface

under the following additional condition:

where

Let M 2 denote the number of rational points ( u 1 , … , u m 2 ) ∈ ( F q * ) m 2 on the variety

under the following additional condition:

where