Counting solutions to a system of quadratic form equations over finite fields

1.   Introduction

The study of equations over finite fields forms a cornerstone of modern algebra, with profound connections to number theory, algebraic geometry, and combinatorics. In particular, the problem of counting the number of solutions to such equations has attracted considerable attention in recent decades. Numerous researchers have explored equations of specific forms, yielding a wealth of results. Early studies, such as ^[5], examined the number of solutions to the equation $a_1X_1^2 + \cdots + a_rX_r^2 = bX_1 \cdots X_r + c$ over a finite field, while ^[2,3,9] investigated certain diagonal equations. More recent contributions include ^[1], which employed the degree reduction matrix to derive an explicit formula for the number of solutions to a polynomial, and ^[6,10,11], which applied Smith's normalized forms to determine the number of solutions to specific equations. For foundational theories and general results concerning equations over finite fields, we refer the reader to ^[7,8]. Nevertheless, determining the exact number of solutions to general polynomial equations or systems of equations remains a highly challenging task.

Let $p$ be a prime and $q$ a power of $p$, and denote by ${\mathbb F}_{q}$ the finite field with $q$ elements. For a positive integer $n$, a quadratic form in $n$ variables over ${\mathbb F}_{q}$ is defined as a homogeneous polynomial of degree $2$ in ${\mathbb F}_{q}[X_1, \ldots, X_n]$, or the zero polynomial. Let $Q(X_1, \dots, X_n)$ be a quadratic form over ${\mathbb F}_{q}$ and $b\in {\mathbb F}_{q}$. We refer to the equation $Q(x_1, \dots, x_n) = b$ as a quadratic form equation over ${\mathbb F}_{q}$ with $n$ unknowns $x_1, \dots, x_n$. It is well known that the number of solutions to a single quadratic form equation over ${\mathbb F}_{q}$ admits a uniform description; see Theorems 6.26, 6.27, and 6.32 in ^[7].

A natural question then arises: what can be said about the number of solutions in ${\mathbb F}_{q}^n$ to systems of multiple quadratic form equations? In fact, this problem was first proposed by Carlitz ^[4], who studied a system consisting of two specific quadratic form equations and derived an explicit formula for the number of solutions in the case of odd characteristic. However, a complete answer to this problem for general systems remains unknown.

Let $m\geq 2$ be an integer, $Q_1(X_1, \dots, X_n), \dots, Q_m(X_1, \dots, X_n)$ be given quadratic forms over ${\mathbb F}_{q}$, and $b_1, \dots, b_m\in {\mathbb F}_{q}$. In this paper, we study the number of solutions in ${\mathbb F}_{q}^n$ to the following system:

Actually, by employing tools from character sums over finite fields, we obtain explicit formulas for the number of solutions in ${\mathbb F}_{q}^n$ to the system (1.1), thereby providing a complete solution to the problem originally raised by Carlitz.

The paper is organized as follows: In Section 2, we present some preliminaries on quadratic forms over finite fields. In Section 3, we derive the main results for the case of odd characteristic. Section 4 is devoted to establishing the corresponding results for even characteristics. Finally, in Section 5, we provide examples to illustrate the application of our main results.

2.   Preliminaries

Let $p$ be a prime, and $q$ be a power of $p$. Denote by ${\mathbb F}_{q}$ the finite field of order $q$. In this section, we present several basic definitions and results concerning quadratic forms over ${\mathbb F}_{q}$.

As usual, for a given column vector $X = (X_1, \dots, X_n)^T$, a linear substitution is defined by $X = CY$, where $C$ is an $n \times n$ matrix over ${\mathbb F}_{q}$, and $Y = (Y_1, \dots, Y_n)^T$ is a new column vector of variables. If $C$ is invertible, the substitution is called a nonsingular linear substitution. Two quadratic forms, $Q_1$ and $Q_2$, are said to be equivalent if there exists a nonsingular linear substitution transforming $Q_1$ into $Q_2$.

Clearly, if $Q_1$ and $Q_2$ are equivalent quadratic forms over ${\mathbb F}_{q}$, then for any $\alpha \in {\mathbb F}_{q}$, the equations $Q_1(x_1, \dots, x_n) = \alpha$ and $Q_2(x_1, \dots, x_n) = \alpha$ have the same number of solutions in ${\mathbb F}_{q}^n$.

For a quadratic form $Q$ over ${\mathbb F}_{q}$, the rank of $Q$, denoted ${\rm{rank}}(Q)$, is defined as the minimal nonnegative integer $r$ such that $Q$ is equivalent to a quadratic form in $r$ variables. It is easy to verify that equivalent quadratic forms have the same rank. In particular, a quadratic form $Q$ in $n$ variables over ${\mathbb F}_{q}$ is said to be nondegenerate if ${\rm{rank}}(Q) = n$.

When $q$ is odd, it is known that any quadratic form $Q(X_1, \ldots, X_n)$ over ${\mathbb F}_{q}$ can be written as

Associated with $Q$ is the $r \times r$ symmetric matrix $A = (a_{ij})$, called the coefficient matrix of $Q$. We denote by $\det(Q)$ the determinant of the coefficient matrix $A$.

A quadratic form $Q$ is called diagonal if its coefficient matrix is diagonal. The following lemma is a standard result:

Lemma 2.1. [7, Theorem 6.21] For $q$ an odd prime power, every quadratic form $Q$ over ${\mathbb F}_{q}$ is equivalent to a diagonal quadratic form with $rank(Q)$ non-zero entries on the main diagonal.

For a quadratic form $Q(X_1, \dots, X_n)$ over ${\mathbb F}_{q}$, we define its diagonal multiplier as $C = c_1 c_2 \cdots c_r$, where $Q$ is equivalent to a diagonal form $c_1 Y_1^2 + \cdots + c_r Y_r^2$ with each $c_i \neq 0$.

For even $q$, we have the parallel result:

Lemma 2.2. [7, Theorem 6.30] Let $q$ be a power of two. Let $Q$ be a quadratic form with rank $n$ over ${\mathbb F}_{q}$. Each of the following statements holds.

(a) If $n$ is odd, then $Q$ is equivalent to

(b) If $n$ is even, then $Q$ is equivalent to

for some $b, d\in {\mathbb F}_{q}$ and $c\in {\mathbb F}_{q}^*$. Moreover, $Q$ is equivalent to

if $b = 0$, or $b\ne 0$ and $\mathit{\text{Tr}}(dbc^{-2}) = 0$, and

if $b\ne 0$ and $\mathit{\text{Tr}}(dbc^{-2}) = 1$. Here, Tr denotes the trace function from ${\mathbb F}_{q}$ to ${\mathbb F}_{2}$.

Let $\chi$ be the canonical additive character of ${\mathbb F}_{q}$ and $\eta$ be the quadratic multiplicative character of ${\mathbb F}_{q}$; that is, $\chi(x) = \exp(2\pi i\text{Tr}(x)/p)$ for any $x\in {\mathbb F}_{q}$, where $\text{Tr}$ is the absolute trace from ${\mathbb F}_{q}$ to its prime field ${\mathbb F}_{p}$, and

Let $g$ be the quadratic Gauss sum over ${\mathbb F}_{q}$ defined by

Its exact value is given by the following lemma.

Lemma 2.3. [7, Theorem 5.15] Let $p$ be an odd prime and $q = p^s$, with $s$ being a positive integer. Then the quadratic Gauss sum $g$ over ${\mathbb F}_{q}$ is given by

where $i$ is the imaginary unit. Consequently, $g^2 = \eta(-1)q$.

For the system of Eq (1.1), let $\mathit{\boldsymbol{b}} = (b_1, \dots, b_m)$, and denote by $N_{\mathit{\boldsymbol{b}}}$ the number of solutions in ${\mathbb F}_{q}^n$ to (1.1). Consider the following associated system:

where $y$ is a new variable distinct from $x_1, \ldots, x_n$. Define $N'_{\mathit{\boldsymbol{b}}}$ as the number of solutions in ${\mathbb F}_{q}^{n+1}$ to the system (2.2). Observe that for each $y \in {\mathbb F}_{q}^*$, the map

induces a bijection from the set of solutions $(x_1, \dots, x_n, y)$ to (2.2) onto the set of solutions to (1.1). Consequently, we have

where $N_{{\bf{0}}}$ denotes the number of solutions in ${\mathbb F}_{q}^n$ to the system (1.1) with $\mathit{\boldsymbol{b}}$ being the zero vector. Thus, to determine $N_{\mathit{\boldsymbol{b}}}$, it suffices to compute the number of solutions in ${\mathbb F}_{q}^n$ to the system of the type

which will be carried out in Section 3 for the case where $q$ is odd and in Section 4 for the case where $q$ is even.

3.   Odd characteristic case

In this section, we investigate the number of solutions to the system (2.4) in ${\mathbb F}_{q}^n$ for odd $q$. We begin by introducing a key lemma.

Lemma 3.1. For an odd prime power $q$, with $\chi$ as the canonical additive character of ${\mathbb F}_{q}$ and $\eta$ as the quadratic multiplicative character of ${\mathbb F}_{q}$, consider a quadratic form $Q(X_1, \dots, X_n)$ in $n$ variables over $\mathbb{F}_q$ with rank $r$. Then

where $\gamma$ is a diagonal multiplier of $Q$, and $g$ is the quadratic Gauss sum defined as in (2.1). In particular, if $r = n$, then

Proof. First, from Lemma 2.1, we know that there exists a nonsingular linear substitution $X = CY$ that transforms $Q$ into the diagonal form

where $c_i\in {\mathbb F}_{q}^*$ for any $1\le i\le r$, and $c_{i} = 0$ for any $i > r$. It then follows that

Let $S = \{x^2: x\in {\mathbb F}_{q}^*\}$. One sees that the map $\sigma: x\mapsto x^2$ from ${\mathbb F}_{q}^*$ onto $S$ is $2$-to-$1$. We also note that

It implies that for any $c\in {\mathbb F}_{q}^*$,

where the last equality holds since $\sum_{y\in {\mathbb F}_{q}}\chi(cy) = 0$. Therefore, we derive that

where $\gamma = c_1\cdots c_r$ is a diagonal multiplier of the quadratic form $Q$. This proves the first part of the lemma. Particularly, if $r = n$, then

Let $A$ be the coefficient matrix of $Q$, and $\Lambda = {\rm{diag}}(c_1, \dots, c_n)$. Then we have $\Lambda = C^{T}AC$. It implies that $\eta(c_1\cdots c_n) = \eta(\det(\Lambda)) = \eta(\det(C)^2\det(A)) = \eta(\det(Q))$. Hence, the second statement follows immediately from (3.1). This completes the proof of Lemma 3.1. □

Before stating the main result, we introduce some notations. Let $q$ be an odd prime power, and let $m\ge 2$ be an integer. Consider $m$ distinct quadratic forms $Q_1(X_1, \dots, X_n)$, $\dots, Q_m(X_1, \dots, X_n)$ in $n$ variables over the finite field ${\mathbb F}_{q}$. Define a polynomial $d(Z_1, \dots, Z_m)$ in ${\mathbb F}_{q}[Z_1, \dots, Z_m]$ as the determinant of the quadratic form $Z_1 Q_1(X_1, \dots, X_n) + \cdots + Z_m Q_m(X_1, \dots, X_n)$ in the variables $X_1, \dots, X_n$. We call $d(Z_1, \dots, Z_m)$ the associated polynomial of the quadratic forms $Q_1, \dots, Q_m$. Clearly, $d(Z_1, \dots, Z_m)$ is either a homogeneous polynomial of degree $n$ or the zero polynomial.

Suppose $d(Z_1, \dots, Z_m)$ has two nonzero linear factors over ${\mathbb F}_{q}$, say

We say that $l_1$ and $l_2$ are equivalent if the vectors $(a_1, \dots, a_m)$ and $(b_1, \dots, b_m)$ are linearly dependent over ${\mathbb F}_{q}$.

Let $L$ denote the set of pairwise inequivalent nonzero linear factors of $d(Z_1, \dots, Z_m)$ over ${\mathbb F}_{q}$, referred to as the linear factor set of $d(Z_1, \dots, Z_m)$. The cardinality of $L$, denoted by $|L| = t$, is called the linear index of the quadratic forms $Q_1, \dots, Q_m$. Clearly, $0 \le t \le n$ if $d(Z_1, \dots, Z_m)$ is not identically zero, and $t = \frac{q^m - 1}{q - 1}$ if $d(Z_1, \dots, Z_m) \equiv 0$.

Define the following set

which is called the set of nontrivial linear zeros of $d(Z_1, \dots, Z_m)$.

We select a maximal subset $\{v_1, \dots, v_s\} \subseteq V$ such that the vectors are pairwise linearly independent over ${\mathbb F}_{q}$; these are referred to as the ${\mathbb F}_{q}$-linearly independent representative set of $V$. For each linearly independent representative $v_i = (z_{i1}, \dots, z_{im}) \in V$, by Lemma 2.1, the quadratic form $z_{i1}Q_1 + \cdots + z_{im}Q_m$ is equivalent to a diagonal quadratic form, which we denote by

where each coefficient $c_{ij} \in \mathbb{F}_q^*$ for all $1 \le j \le r_i$. We denote $C_i = c_{i1} \cdots c_{ir_i}$ and refer to it as a diagonal multiplier of $v_i$. The integer $r_i$ is called the rank of $v_i$.

Now we can report the main result of this section as follows.

Theorem 3.1. For an odd prime power $q$ and an integer $m \geq 2$, let $Q_1, \dots, Q_m$ be distinct quadratic forms in $n$ variables $X_1, \dots, X_n$ over ${\mathbb F}_{q}$, with associated polynomial $d(Z_1, \dots, Z_m)$. Define $V$ as the set of nontrivial linear zeros of $d(Z_1, \dots, Z_m)$. Denote by $N$ the number of solutions in ${\mathbb F}_{q}^n$ to the system (2.4). The following statements hold:

(a) If $V$ is nonempty with an ${\mathbb F}_{q}$-linearly independent representative set $\{v_1, \dots, v_s\}$, then

where

$r_i$ and $C_i$ are the rank and the diagonal multiplier of $v_i$, respectively.

(b) If $V$ is empty, then

where $\mathcal{V}$ is defined as in (a).

Proof. Let $\chi$ be the canonical character of ${\mathbb F}_{q}$. First, using the orthogonality of characters, we express $N$ as

For $z = (z_1, \dots, z_m)\in {\mathbb F}_{q}^m$, define a sum

Clearly, $S({\bf{0}}) = q^n$, where ${\bf{0}}$ denotes the zero vector in ${\mathbb F}_{q}^m$. Let $d(Z_1, \dots, Z_m)$ be the associated polynomial of $Q_1, \dots, Q_m$, and let $V$ be the set of nontrivial linear zeros of $d(Z_1, \dots, Z_m)$. Then we have

Suppose that $V$ is nonempty with an ${\mathbb F}_{q}$-linearly independent representative set $\{v_1, \dots, v_s\}$. For each $v_i$, let $C_i$ and $r_i$ be a diagonal multiplier and the rank of $v_i$, respectively. It follows that

Note that $\lambda v_i$ has a diagonal multiplier $\lambda^{r_i}C_i$ for any $\lambda\in {\mathbb F}_{q}^*$. Thus, by using Lemma 3.1, we obtain

where the last equality holds since

Now we turn attention to the last sum on the right-hand side of (3.3). Note that the quadratic form

is of rank $n$ for any $(z_1, \dots, z_m)\in {\mathbb F}_{q}^m\setminus V$, and recall that $d(Z_1, \dots, Z_m)$ is the associated polynomial of $Q_1, \dots, Q_m$. It follows from Lemma 3.1 that

since $\eta(0) = 0$. In particular, for odd $n$, consider the sum

Let $w\in {\mathbb F}_{q}^*$ be a non-square element. Then

Since $n$ is odd and $\eta(w) = -1$, we have $\eta(w^n) = -1$, so $\mathcal{S} = -\mathcal{S}$, implying $\mathcal{S} = 0$. By substituting (3.4) and (3.5) into (3.3), combining with (3.2), and applying Lemma 2.3, we obtain the first statement of this theorem.

If $V$ is empty, the first sum on the right-hand side of (3.3) vanishes, and the second statement of this lemma follows similarly.

This completes the proof of Theorem 3.1. □

4.   Even characteristic case

In this section, we analyze the number of solutions to the system (2.4) in $\mathbb{F}_q^n$ for even $q$. We start with a useful lemma.

Lemma 4.1. Suppose $q$ is a power of two. Take $\chi$ as the canonical additive character of ${\mathbb F}_{q}$, and let $Q(X_1, \dots, X_n)$ be a quadratic form in $n$ variables over ${\mathbb F}_{q}$ with rank $r$. Then

where $\delta = 1$ if $Q(X_1, \dots, X_n)$ is equivalent to $Y_1Y_2+Y_3Y_4+\cdots+Y_{r-1}Y_r$, and $\delta = -1$ otherwise.

Proof. First, we let $r$ be odd. Using Lemma 2.2(a), we can transform the quadratic form $Q(X_1, \dots, X_n)$ into

by a nonsingular linear substitution. Then

Note that $y^2$ is a permutation polynomial of ${\mathbb F}_{q}$ since $q$ is even. It implies that

Hence, we obtain

Now, let $r$ be even. Similarly, from Lemma 2.2(b), we know that $Q(X_1, \dots, X_n)$ is equivalent to

or

for some $a\in {\mathbb F}_{q}$ with ${\rm{Tr}}(a) = 1$. For the first case, we have

And we compute that

Substituting (4.2) into (4.1) yields the desired result of this case. For the latter case, we have

We need to evaluate the last sum of (4.3). On the one hand,

On the other hand, we note that ${\rm{Tr}}(a) = 1$, and then $t^2+t+a\ne 0$ for any $t\in {\mathbb F}_{q}$. It implies that

Therefore, by substituting (4.2) and (4.4) into (4.3), we arrive at the final result of this case.

This proves Lemma 4.1. □

We now state the main result of this section.

Theorem 4.1. Suppose $q$ is a power of two and $m \geq 2$ is an integer. Consider $Q_1, \dots, Q_m$ as distinct quadratic forms in $n$ variables $X_1, \dots, X_n$ over ${\mathbb F}_{q}$. For a vector $z = (z_1, \dots, z_m) \in {\mathbb F}_{q}^m$, define the quadratic form $Q(z)$ in variables $X_1, \dots, X_n$ over ${\mathbb F}_{q}$ by $Q(z) = z_1 Q_1 + \cdots + z_m Q_m$. For each $0 \leq r \leq n$, define $T_r$ as the set of vectors $z \in {\mathbb F}_{q}^m$ such that $Q(z)$ has rank $r$, and define $T_r'$ as the subset of $z \in {\mathbb F}_{q}^m$ such that $Q(z)$ has rank $r$ and is equivalent to $Y_1 Y_2 + Y_3 Y_4 + \cdots + Y_{r-1} Y_r$. Then the number $N$ of solutions in ${\mathbb F}_{q}^n$ to the system (2.4) is

where $|S|$ represents the number of elements in the set $S$.

Proof. Let $\chi$ be the canonical character of ${\mathbb F}_{q}$. From the definition of $T_r$, we can partition ${\mathbb F}_{q}^m$ into ${\mathbb F}_{q}^m = \bigcup_{r = 0}^nT_r$. For $z = (z_1, \dots, z_m)\in {\mathbb F}_{q}^m$, let $Q(z) = z_1Q_1+\cdots+z_mQ_m$. By (3.2), we then have

Applying Lemma 4.1 into (4.5), we obtain

where $\delta_{r, z} = 1$, if $Q(z)$ is equivalent to $Y_1Y_2+Y_3Y_4+\cdots+Y_{r-1}Y_r$, and otherwise $\delta_{r, z} = -1$. Then $N$ can be simplified to

as desired. Theorem 4.1 is proved. □

5.   Concluding remarks and examples

In this paper, we investigate the number of solutions to systems of quadratic form equations over finite fields, deriving explicit formulas for the solution counts, thereby addressing one of Carlitz's problems; see Theorems 3.1 and 4.1. Our results, while not always straightforward, offer a novel algorithm for computing the number of solutions. For instance, in Theorem 3.1, one must first determine the set $V$ of nontrivial linear zeros of the associated polynomial $d(Z_1, \dots, Z_m)$ for the quadratic forms $Q_1, \dots, Q_m$, then identify a linearly independent representative subset $\{v_1, \dots, v_s\} \subseteq V$, and compute the rank and diagonal multiplier of each $v_i$. The desired result is then obtained using our formula. Similarly, for Theorem 4.1, one must determine the sizes of the sets $T_{2r}$ and $T'_{2r}$, which depend on the quadratic forms $Q_1, \dots, Q_m$. However, deriving closed-form formulas for $|T_{2r}|$ and $|T'_{2r}|$ remains a significant challenge.

Below, we present several examples to illustrate the application of our theorems.

5.1.   Example 5.1

Let $\mathbb{F}_9 = \mathbb{F}_3(a)$, where $a$ is a defining element of $\mathbb{F}_9$ over $\mathbb{F}_3$ satisfying $a^2 = a + 1$. We apply Theorem 3.1 to determine the number of solutions to the following system over $\mathbb{F}_9$:

To compute the number of solutions to system (5.1), we consider two auxiliary systems:

with $N_1$ solutions in $\mathbb{F}_9^6$, and

with $N_2$ solutions in $\mathbb{F}_9^5$.

5.1.1.   Solutions to system (5.2)

To determine $N_1$, we proceed as follows:

Step 1. Compute the associated polynomial and factorize it over $\mathbb{F}_9$:

Step 2. Select linearly independent representatives of the linear zero set of $d(Z_1, Z_2)$, given by $v_1 = (0, 1)$, $v_2 = (1, -1)$, and $v_3 = (1, 1)$.

Step 3. For each $v_i$, compute the rank and the diagonal multiplier of the corresponding quadratic form in the standard way:

● For $v_1 = (0, 1)$, the quadratic form $Q_2'$ has the equivalent diagonal form $Y_1^2 + Y_2^2 + Y_3^2 + Y_4^2 - Y_5^2$, with rank 5 and diagonal multiplier $-1$.

● For $v_2 = (1, -1)$, the quadratic form $Q_1' - Q_2'$ has the equivalent diagonal form $Y_1^2 + Y_2^2 + Y_3^2 - Y_4^2 - Y_5^2$, with rank 5 and diagonal multiplier $1$.

● For $v_3 = (1, 1)$, the quadratic form $Q_1' + Q_2'$ has the equivalent diagonal form $Y_1^2 + Y_2^2 - Y_3^2 - Y_4^2 - Y_5^2$, with rank 5 and diagonal multiplier $-1$.

Step 4. Compute $\mathcal{W}$, obtaining $\mathcal{W} = 0$.

Step 5. Calculate $\mathcal{V}$ as:

where $\eta$ is the quadratic character of $\mathbb{F}_9$.

Step 6. Substitute the computed values into Theorem 3.1, yielding

solutions for system (5.2).

5.1.2.   Solutions to system (5.3)

To determine $N_2$, we follow a similar procedure:

Step 1. Compute the associated polynomial and factorize it over $\mathbb{F}_9$:

Step 2. Select linearly independent representatives of the linear zero set of $d(Z_1, Z_2)$, given by $v_1 = (0, 1)$ and $v_2 = (1, -1)$.

Step 3. For each $v_i$, compute the rank and multiplier of the corresponding quadratic form:

● For $v_1 = (0, 1)$, the quadratic form $Q_2$ has the equivalent diagonal form $Y_1^2 + Y_2^2 + Y_3^2 + 2 Y_4^2$, with rank 4 and diagonal multiplier $-1$.

● For $v_2 = (1, -1)$, the quadratic form $Q_1 - Q_2$ has the equivalent diagonal form $Y_1^2 + Y_2^2 + 2 Y_3^2 + 2 Y_4^2$, with rank 4 and diagonal multiplier $1$.

Step 4. Compute $\mathcal{W}$, obtaining

Step 5. The calculation of $\mathcal{V}$ is not required, as $n = 5$ is odd.

Step 6. Substitute the computed values into Theorem 3.1, yielding

solutions for system (5.3).

5.1.3.   Solutions to system (5.1)

Using the relationship given by (2.3), the number of solutions to system (5.1) is

5.2.   Example 5.2

Let $\mathbb{F}_4 = \{0, 1, w, w^2\}$, where $w$ is a primitive cube root of unity. We apply Theorem 4.1 to determine the number $N$ of solutions in $\mathbb{F}_4^7$ to the following system over $\mathbb{F}_4$:

To compute the number of solutions to (5.4), we proceed as follows:

Step 1. For each $(z_1, z_2) \in \mathbb{F}_4^2$, define the quadratic form:

Step 2. Classify $Q(z_1, z_2)$ by its rank. For each $(z_1, z_2) \in \mathbb{F}_4^2$, compute the rank of the quadratic form $Q(z_1, z_2)$. In finite fields of even characteristic, the rank of a quadratic form can be determined using a standard method, as described in [7, Lemma 6.29 and Theorem 6.30]. Specifically, a non-singular linear transformation can be applied to convert the quadratic form into a canonical form, and the rank is given by the number of variables with non-zero coefficients in this canonical form.

Step 3. Determine the sets $T_r$ and $T_r'$ as defined in Theorem 4.1. Through computation, we obtain:

with all other $T_r$ and $T_r'$ being empty.

Step 4. Substitute the computed values of $|T_r|$ and $|T_r'|$ into Theorem 4.1, yielding the number of solutions $N = 1024$ for system (5.4).

Author contributions

Xiaodie Luo and Kaimin Cheng: Conceptualization, Methodology, Validation, Writing-original draft, Writing-review & editing. All authors contributed equally to this work.

Use of Generative-AI tools declaration

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

Acknowledgment

The authors would like to thank the anonymous referees for their insightful and valuable comments, which significantly enhanced the paper's presentation. The second author thanks Arne Winterhof for his hospitality and support during the second author's visit to RICAM.

This research was partially supported by the China Scholarship Council Fund (Grant No. 202301010002) and the Scientific Research Innovation Team Project of China West Normal University (Grant No. KCXTD2024-7).

Conflict of interest

The authors declare that they have no conflicts of interest.