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

1.   Introduction

Throughout this paper, $p$ will always denote an odd prime, $\mathbb Z^+$ and $\mathbb F_q$ denote the set of positive integers and the finite field having $q = p^\eta$ elements, respectively, where $\eta\in \mathbb Z^+$. Then $\mathbb F_q^*: = \mathbb F_q\setminus\{0\}$ forms a group under the multiplicative operation. For any finite set $\mathcal S$, $|\mathcal S|$ means its cardinality. Let $\lambda, n\in \mathbb Z^+$ and $\langle \lambda\rangle$ be the set of the first $\lambda$ positive integers. Let $x_1, ..., x_{n-1}$ and $x_n$ be $n$ indeterminates in $\mathbb F_q$, and for brevity, let ${\bf x} = (x_1, ..., x_n)$. Let $f_1({\bf x}), ..., f_\lambda({\bf x})$ be the system of $n$-variable polynomials over $\mathbb F_q$, and we denote by $V(f_1, ..., f_\lambda) = V(f_1({\bf x}), ..., f_\lambda({\bf x}))$ the affine variety determined by the vanishing of these polynomials. Define

When $\lambda = 1$, one writes $N(V) = N(f)$. Finding an accurate formula for $N(V)$ is a common and significant subject. However, such a problem is hard in general. In the past 70 years, many mathematicians were devoted to this subject and made much vital progress (see ^{[1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]}).

In 1997, the number $N(f)$ of rational points over $\mathbb F_q$ on the following affine hypersurface

was investigated by Sun ^[18]. Besides, the accurate formula for the number $N(f)$ of rational points was found in ^[18]:

provided $s = n$ and $\gcd(\det(e_{ij}), q-1) = 1$, where $A(𝓈): = (q-1)^{𝓈}-(-1)^{𝓈}, \ \forall 𝓈\in \mathbb Z^+$. Eight years later, the result of ^[18] was successfully extended by Wang and Sun ^[21]. Actually, they attained a formula for the number of $(x_1, ...x_{n_2})\in \mathbb F_q^{n_2}$ on the following hypersurface

with $d_{ij}\in \mathbb Z^+, a_i\in \mathbb F_q^*, 1\le i, j\le n_2$.

In 2015, Hu, Hong and Zhao ^[9] gave a uniform generalization to the results of ^[20,21]. Actually, they used the Smith normal form to deduce an accurate formula for $N(f)$ of $(x_1, ...x_{n_t})\in \mathbb F_q^{n_t}$ on the hypersurface over $\mathbb F_q$ defined by

where the integers $t > 0$, $0 = r_0 < r_1 < r_2 < ... < r_t$, $1\le n_1 < n_2 < ... < n_t$, $b\in \mathbb F_q$, $a_i\in\mathbb F_q^*$ and $e_{ij}\in \mathbb Z^+$, $i\in\langle r_t\rangle$, $j\in\langle n_t\rangle$. Under some restrictions on $f$, a little bit simple formula about the number of rational points on the hypersurface (1.1) was given in ^[13]. One notices that the result of ^[9] was extended by Hu and Zhao ^[11] from the hypersurface case to certain algebraic variety case.

An open problem was raised at the end of [9, Section 3]. For the case of the variety consisting of two hypersurfaces, Hu, Qin and Zhao ^[10] and Zhu and Hong ^[27] obtained some partial answers to this problem. In other words, Hu, Qin and Zhao ^[10] gave an explicit formula for $N(V(f_1, f_2))$, where

with $1\le r_1 < r_2, 1\le r_3 < r_4, 1\le n_1 < n_2, 1\le n_3 < n_4$, $e^{(1)}_{i, j}, \ e^{(2)}_{i', j'}\in \mathbb Z^+$, $b_1, b_2\in\mathbb F_q$, and $a_{1i}, a_{2i'}\in \mathbb F_q^*$, $i\in\langle r_2\rangle$, $i'\in\langle r_4\rangle$, $j\in\langle n_2\rangle$, $j'\in\langle n_4\rangle$. Zhu and Hong ^[27] used and developed the techniques in ^[9] to get an exact formula for the number of rational points on $V = V(f_1, f_2)$ over $\mathbb{F}_q$ with

where $b_i\in \mathbb F_q, i = 1, 2, \ t\in \mathbb Z^+$, $0 = n_0 < n_1 < n_2 < ... < n_t$, $n_{k-1} < n\le n_k$ for some $k\in\langle t\rangle$, $0 = r_0 < r_1 < r_2 < ... < r_t$, $a^{(1)}_i, a^{(2)}_{i'}\in \mathbb F_q^*, i\in\langle r\rangle, i'\in\langle r_t\rangle$, $e^{(1)}_{i, j}, \ e^{(2)}_{i', j'}\in \mathbb Z^+$, $j\in\langle n\rangle$, $j'\in\langle n_t\rangle$.

Inspired by the works of ^[9,18,21,27], we consider in this paper the question of counting rational points on the variety $V(f_1, f_2)$ with

and the variety $V(f_1, f_2, f_3)$ with

where $d_{ij}, e_{i'j'}\in \mathbb Z^+, a_i, c_{i'}\in \mathbb F_q^*, i\in\langle s+t\rangle, j\in\langle n\rangle, i'\in\langle r+k+w\rangle, j'\in\langle m\rangle,$ and $b_1, b_2, l_1, l_2, l_3\in \mathbb F_q$. Let

with $d_{ij}, i\in \langle s+t\rangle, j\in \langle n\rangle$ being given as in (1.3), and let

with $e_{i'j'}, i'\in\langle r+k+w\rangle, j'\in\langle m\rangle$ being given as in (1.4).

From ^[12], it guarantees the existences of unimodular matrices $U_2$ and $V_2$ with the property

where

with $g^{(E_2)}_1, ..., g^{(E_2)}_{v'}\in \mathbb Z^+$ and $g^{(E_2)}_1|...|g^{(E_2)}_{v'}$. The diagonal matrix on the right side of (1.7) is called Smith normal form of $E_2$, and abbreviated as ${\rm SNF}(E_2)$. That is,

Fix $\alpha \in \mathbb F_q^*$ as a primitive element of $\mathbb F_q$, then for any $\beta \in \mathbb F_q^*$, one can find a unique integer $\gamma\in [1, q-1]$ with $\beta = \alpha^{\gamma}$, and such an integer $\gamma$ is said to be the index of $\beta$ on the basis $\alpha$. We write ${\rm ind}_{\alpha}\beta: = \gamma$.

Consider the variety defined by

where $c_{i'}, l_1, l_2, l_3, i'\in\langle r+k+w\rangle$ are given as in (1.4). Let $\mathscr N$ denote the number of rational points $(𝓋_1, ..., 𝓋_{r+k+w}) \in (\mathbb F_q^*)^{r+k+w}$ on (1.8) satisfying

where

We can now state our main results.

Theorem 1.1. Let $V$ be the variety defined by (1.3). If $s+t = n$ and $\gcd(q-1, \det(E_1)) = 1$, then

Theorem 1.2. Let $V$ be the variety defined by (1.4). Then

where $R: = (q-1)^{m-v'}\prod\limits_{j = 1}^{v'}\gcd (q-1, g^{(E_2)}_j)$.

From Theorem 1.2, one can derive the third main result of this paper as follows.

Theorem 1.3. Let $V$ denote the affine variety defined by (1.4). If $r+k+w = m$ and $\gcd(q-1, \det(E_2)) = 1$, then

Obviously, Theorems 1.1 to 1.3 also give a partial answer to the open problem proposed at the end of [9, Section 3].

In Section 2, in order to prove Theorems 1.1 to 1.3, we give several auxiliary results. Then in Section 3, one presents the details of the proofs of Theorems 1.1 to 1.3. Finally, four examples are provided in Section 4.

2.   Auxiliary results

In this section, we present several preliminary results which are needed in the proofs of Theorems 1.1 to 1.3. We begin with a result due to Zhu and Hong ^[27].

Lemma 2.1. [27, Lemma 2.6] Let $c_{ij}\in \mathbb F_q^*$, $i\in\langle𝓂\rangle$, $j\in\langle 𝓀_i\rangle$, $c_1, ..., c_{𝓂}\in \mathbb F_q$. Let $N(c_1, ..., c_{𝓂})$ stand for the number of $(u_{11}, ..., u_{1𝓀_1}, ..., u_{𝓂1}, ..., u_{𝓂𝓀_{𝓂}})\in (\mathbb F_q^*)^{𝓀_1+...+𝓀_{𝓂}}$ such that

Then

Lemma 2.2. Let $a_i\in \mathbb F_q^*$, $i\in\langle s+t\rangle$, $b_1, b_2\in \mathbb F_q$. Let $N(b_1, b_2)$ denote the number of $(𝓊_1, ..., 𝓊_{s+t})\in (\mathbb F_q^*)^{s+t}$ with

Then

Proof. The result follows immediately from Lemma 2.1. □

Lemma 2.3. Let $c_i\in \mathbb F_q^*$ for all $i\in\langle r+k+w\rangle$ and let $l_1, l_2, l_3\in \mathbb F_q$. Let $N(l_1, l_2, l_3)$ denote the number of $(𝓋_1, ..., 𝓋_{r+k+w})\in (\mathbb F_q^*)^{r+k+w}$ satisfying (1.8). Then

Proof. This follows immediately from Lemma 2.1. □

Reference [12] tells us that by using elementary transformation, we can readily find unimodular matrices $U_1$ and $V_1$ with the property

where $E_1$ is given as in (1.5),

with $g^{(E_1)}_1, ..., g^{(E_1)}_v\in \mathbb Z^+$ and $g^{(E_1)}_1|...|g^{(E_1)}_v$. Let $\mathscr M$ represent the number of $(𝓊_1, ..., 𝓊_{s+t})\in (\mathbb F_q^*)^{s+t}$ on (2.1) under the following additional restrictions:

where

As a special case of [27, Theorem 1.2], one has the following result.

Lemma 2.4. Let $V$ be the variety (1.3). Then

Let $g_{ij}, 𝒷_i (i\in\langle 𝓁\rangle, j\in\langle 𝓊\rangle)$ and ${𝒶}$ be integers. Let $Y = (y_1, ..., y_{𝓊})^{\mathrm{T}}$ and $\mathscr B = (𝒷_1, ..., 𝒷_𝓁)^{\mathrm{T}}$. Then one forms an $𝓁\times 𝓊$ matrix $\mathscr G = (g_{ij})$ and the following system of congruences:

From ^[12], one can use elementary transformation of matrices to find unimodular matrices $\mathcal U$ and $\mathcal V$ with the property

where $\mathscr D: = {\rm diag}(d_1, ..., d_\tau)$ with $d_i\in\mathbb Z^+, i\in\langle \tau\rangle$ and $d_i|d_{i+1}, i\in\langle\tau-1\rangle$.

Lemma 2.5. [9, Lemma 2.3] Let $\mathscr B^{'} = (𝒷_1^{'}, ..., 𝒷_𝓁^{'})^{\mathrm{T}} = \mathcal U\mathscr B,$ then a necessary and sufficient condition for the system (2.6) of linear congruences to have a solution is $\gcd(𝒶, d_i)|𝒷_i^{'}$ for all $i\in\langle \tau \rangle$ and $𝒶|𝒷_i^{'}$ for all $i\in\langle𝓁\rangle\setminus\langle\tau\rangle$. In addition, the number of solutions $(y_1, ..., y_𝓊)^{\mathrm{T}}$ of (2.6) equals $𝒶^{{𝓊}-\tau}\prod\limits_{i = 1}^{\tau}\gcd(𝒶, d_i)$.

Lemma 2.6. Let $r, k$ and $w$ be positive integers. Then

Proof. First of all, for any given $(𝓋_1, ..., 𝓋_{r+k+w}) \in (\mathbb F_q^*)^{r+k+w}$ satisfying (1.8), we have the system of congruences:

then

However, Lemma 2.5 tells us that the necessary and sufficient condition for (2.8) to have a solution is that (1.9) holds. In addition, if (2.8) has a solution, then the number of the $m$-tuples $({\rm ind}_\alpha(x_1), ..., {\rm ind}_\alpha(x_m))\in \langle q-1\rangle^m$ satisfying (2.8) is equal to

Namely, if (1.9) is satisfied, then

Thus, the left hand side of (2.7) is equal to

However,

Hence putting (2.10) into (2.9) gives us the wanted result (2.7). □

3.   Proofs of Theorems 1.1 to 1.3

In this section, we present the proofs of Theorems 1.1 to 1.3. We begin with the proof of Theorem 1.1.

Proof of Theorem 1.1. Taking determinants on both sides of (2.4), we can deduce that

Since $\det(U_1) = \pm 1$ and $\det(V_1) = \pm 1$, the condition

implies that

So (2.5) holds.

Further, by Lemma 2.2, one has

with $N(b_1, b_2)$ being given as in (2.2). It follows from Lemma 2.4 that

Thus, putting (3.1) and (3.2) together gives the expected result (1.10).

This completes the proof of Theorem 1.1. □

Proof of Theorem 1.2. It is clear that

One defines the set $T(l_1, l_2, l_3)$ of $\mathbb F_q$-rational points as follows:

Substituting (3.4) into (3.3) yields

Define the set $T(0)$ by $T(0): = \varnothing$ if $l_1, l_2$ and $l_3$ are not all zero, and if $l_1 = l_2 = l_3 = 0$, then $T(0)$ consists of the zero vector of dimension $r+k+w$. For any integer $\rho$ with $1\le\rho\le r+k+w$, one defines the set $T(\rho)$ to be the subset of $T(l_1, l_2, l_3)$ consisting of $(𝓋_1, ..., 𝓋_{r+k+w}) \in\mathbb F_q^{r+k+w}$ with exactly $\rho$ nonzero components. Noticing that $𝓋_1, ..., 𝓋_{r+k+w}$ are simultaneously zero, or simultaneously nonzero, one has $T(\rho) = \varnothing$ when $0 < \rho < r+k+w$. Hence,

Now, applying Lemma 2.6, we have

It readily follows that if at least one of $l_1, l_2$ and $l_3$ is nonzero, then $T(0) = \varnothing$, and so by (3.5) to (3.7), one has

If $l_1 = l_2 = l_3 = 0$, then by using (3.5) to (3.7), we derive that

as expected.

This finishes the proof of Theorem 1.2. □

Proof of Theorem 1.3. Taking determinants on both sides of (1.7), one has

Because $\det(U_2) = \pm 1$ and $\det(V_2) = \pm 1$, the condition $\gcd(q-1, \det(E_2)) = 1$ guarantees that

This ensures that (1.9) is satisfied.

Noting that

with $N(l_1, l_2, l_3)$ being given as in (2.3), it follows from (1.11) that

Therefore, by the identities (3.8) and (3.9), the desired result (1.12) follows immediately.

This concludes the proof of Theorem 1.3. □

4.   Examples

In this section, we give four examples to demonstrate the validity of Theorems 1.1 to 1.3.

Example 4.1. We calculate the number $N(V)$ of rational points on the variety

over $\mathbb F_{11}$.

Clearly, we have

and

Since $\det(E_1) = 9$, one derives that $\gcd(q-1, \det(E_1)) = 1$. By Theorem 1.1, we can calculate and obtain that

Example 4.2. We compute the number $N(V)$ of rational points on the variety

over $\mathbb F_7$.

Evidently, we have

and

Hence, $\det(E_2) = -22$, one observes that $\gcd(q-1, \det(E_2))\ne 1$.

By using Maple, we can find two unimodular matrices

and

such that

Thus,

Still using Maple, we compute and get that the number $\mathscr N$ of vectors $(𝓋_1, ..., 𝓋_6)\in (\mathbb F_q^*)^6$ with

under the extra restriction (1.9) is equal to 108. Thus, by Theorem 1.2, we have

Example 4.3. We compute the number $N(V)$ of rational points on the variety

over $\mathbb F_{13}$.

Obviously, we have

and

Since $\det(E_2) = 4387 = 41\times 107$, we deduce that $\gcd(q-1, \det(E_2)) = 1$. By Theorem 1.3, we compute and obtain that

Example 4.4. We calculate the number $N(V)$ of rational points on the variety

over $\mathbb F_{11}$.

It is clear that

and

Thus, $\det(E_2) = 957$ which infers that $\gcd(q-1, \det(E_2)) = 1$. Therefore, by employing Theorem 1.3, we compute and obtain that

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The research of M. Li was supported partially by National Science Foundation of China (Grant No. 12071375).

Conflict of interest

The authors declare that there is no conflict of interest in this paper.