The number of solutions of cubic diagonal equations over finite fields

1.   Introduction and statement of main result

Let $p$ be a prime, $k$ be a positive integer, $q = p^k$. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*$ represent the nonzero elements of $\mathbb{F}_{q}$. Let $f(x_1, \cdots, x_s)\in\mathbb{F}_q[x_1, \cdots, x_s]$ be a polynomial over $\mathbb{F}_q$ with $s$ variables. A solution (or a rational point) of $f(x_1, \cdots, x_s)$ over $\mathbb{F}_q$ is an $s$-tuple $(c_1, \cdots, c_s)\in\mathbb{F}_{q}^s$ such that $f(c_1, \cdots, c_s) = 0$. Denote by

the number of solutions of $f(x_1, \cdots, x_s) = 0$ over $\mathbb{F}_{q}$.

It is one of the central problems to study the number $N(f, q)$ of rational points over finite fields. From ^[14,15] we know that there exists an explicit formula for $N(f, q)$ with degree $\text{deg}(f) \leq2$. But generally speaking, it is much difficult to give an explicit formula for $N(f, q)$. Finding the explicit formula for $N(f, q)$ under certain conditions has attracted many researchers for many years (See, for instance, ^{[1,3,4,5,6,7,8,9,10,11,12,13,16,17,18,20,21,22,23,24,25,26,27]}).

Let $k_1, \dots, k_s$ be positive integers. A diagonal equation is an equation of the form

with coefficients $a_1, \dots, a_s\in\mathbb{F}_{q}^*$ and $c\in\mathbb{F}_{q}$. The special case where all the $k_i$'s are equal has extensively been studied (see, for example, ^{[7,8,9,10,11,12,17,18,21,22,23,26,27]}).

In 1977, S. Chowla et al. (^[7]) investigated a problem about the number of solutions of an equation

over field $\mathbb{F}_{p}$, where $p$ is a prime with $p\equiv1(\rm{mod} \ 3)$. In 1979, Myerson ^[17] extended the result in ^[7] to the field $\mathbb{F}_{q}$ and also studied the number of solutions of the equation

over $\mathbb{F}_{q}$. When $q = p^{2t}$ with $p^r\equiv -1({\rm{mod}} \ d)$ for a divisor $r$ of $t$ and $d\mid (q-1)$, Wolfmann ^[22] gave an explicit formula of the number of solutions of the equation

over $\mathbb{F}_{q}$ in 1992, where $a_1, a_2, \dots, a_s\in \mathbb{F}_{q}^*$ and $c\in \mathbb{F}_{q}$. In 2018, Zhang and Hu ^[23] determined the number of solutions of the equation

over $\mathbb{F}_{p}$, with $c\in\mathbb{F}_{p}^*$ and $p\equiv1(\rm{mod} \ 3)$.

In 2020, J. Zhao et al. ^[26,27] investigated the number of solutions of the equations

over $\mathbb{F}_{q}$, with $c\in\mathbb{F}_{q}^*$.

For any $c\in\mathbb{F}_{q}$, let $A_n(c)$ and $B_n(c)$ denote the number of solutions of the equations $x_1^3 +x_2^3 +\cdots+ x_n^3 = c$ and $x_1^3 +x_2^3 + \cdots+ x_n^3+ cx_{n+1}^3 = 0$ over $\mathbb{F}_q$ respectively. In 2021, by using the generator of $\mathbb{F}_{q}^*$, Hong and Zhu ^[10] gave the generating functions $\sum\limits_{n = 1}^{\infty}A_n(c)x^n$ and $\sum\limits_{n = 1}^{\infty}B_n(c)x^n$. In 2022, W. Ge et al. ^[9] studied these two generating functions in a different way. Moreover, formulas of the number of solutions of equation $a_1x_1^3+a_2x_2^3 = c$ and $a_1x_1^3+a_2x_2^3+a_3x_3^3 = 0$ were also presented in ^[9].

In this paper, we consider the problem of finding the number of solutions of the diagonal cubic equation

over $\mathbb{F}_q$, where $q = p^k$ and $a_1, a_2, \dots, a_s\in\mathbb{F}_q^*$ and $c\in\mathbb{F}_q$.

If $p = 3$ and $k$ is an integer, or $p\equiv2(\rm{mod} \ 3)$ and $k$ is an odd integer, then $\gcd(3, q-1) = 1$. It follows that (see ^[14] pp.105)

with $a_1, a_2, \dots, a_s\in\mathbb{F}_q^*$, and $c\in\mathbb{F}_q$.

If $p\equiv2 (\rm{mod} \ 3)$ and $k$ is an even integer, Hua and Vandiver ^[13] studied the number of solutions of some trinomial equations over $\mathbb{F}_q$ and Wolfmann ^[22] also got the number of solutions of certain diagonal equations over $\mathbb{F}_q$. The following result can be deduced from Theorem 1 of ^[22].

Theorem 1.1. Let $p\equiv2(\rm{mod} \ 3)$ be a prime, $k$ an even integer, $q = p^k$, $n = \frac{q-1}{3}$, $s\ge 2$ and $c\in\mathbb{F}_q$. Let $\alpha$ be a primitive element of $\mathbb{F}_q$. Denote by $N$ the number of solutions of the equation

over $\mathbb{F}_q$. Then

if $c = 0$, and

if $c\neq0$, where $v(j)$ is the number of $i$, $1\leq i\leq s$, such that $(\alpha^j)^n a_i^n = (-1)^{k(p+1)/6}$; $\theta(c)$ is the number of $i$, $1\leq i\leq s$, such that $a_i^n = (-c)^n$ and $\tau(j)$ is the number of $i$, $1\leq i\leq s$, such that $a_i^n = (\alpha^j)^n$.

However, the explicit formula for $N(a_1x_1^3+a_2x_2^3+\cdots+a_sx_s^3 = c)$ is still unknown when $p\equiv1(\rm{mod} \ 3)$. In this paper, we solve this problem by using Jacobi sums and an analog of Hasse-Davenport theorem. We give an explicit formula for the number of solutions of diagonal cubic equations

and

over $\mathbb{F}_q$, with $a_1, a_2, b_1, b_2, b_3\in\mathbb{F}_q^*$, $c\in\mathbb{F}_q$ and the characteristic $p\equiv 1(\rm{mod}\ 3)$. Note that our approach, which applies Jacobi sums, is not the same as that of Ge et al. ^[9] and Hong and Zhu ^[10] which mainly applies Gauss sums. The case with arbitrary $s\ge 4$ variables can be deduced from the reduction formula for Jacobi sums. But we omit the tedious details here.

Let $\alpha \in\mathbb{F}^*_q$ be a fixed primitive element of $\mathbb{F}_q$. For any $\beta\in\mathbb{F}^*_q$, there exists exactly one integer $r\in [1, q-1]$ such that $\beta = \alpha^r$. Such an integer $r$ is called the index of $\beta$ with the primitive element $\alpha$, and denoted by $\text{ind}_\alpha\beta: = r$. For any nonzero integer $m$ and prime number $p$, we define $\nu_p(m)$ as the greatest integer $t$ such that $p^t$ divides $m$. Then $\nu_p(m)$ is a nonnegative integer, and $\nu_p(m)\geq$ 1 if and only if $p$ divides $m$.

The results of this paper are stated as follows.

Theorem 1.2. Let $k$ be a positive integer and $q = p^k$ with the prime $p\equiv 1(\rm{mod}\ 3)$. Let $\alpha$ be a primitive element of $\mathbb{F}_q$. Denote by $N_1$ the number of solutions of the equation $a_1x_1^3+a_2x_2^3 = c$ over $\mathbb{F}_q$. Then

if $c = 0$, and

if $c\neq0$, where

and the integers $u$ and $v$ are uniquely determined such that

Theorem 1.3. Let $k$ be a positive integer and $q = p^k$ with the prime $p\equiv 1(\rm{mod}\ 3)$. Let $\alpha$ be a primitive element of $\mathbb{F}_q$. Denote by $N_2$ the number of solutions of the equation $b_1x_1^3+b_2x_2^3+b_3x_3^3 = c$ over $\mathbb{F}_q$. Then

if $c = 0$, and

if $c\neq0$, where $E(u, v, k)$, $O(u, v, k)$, $u$, $v$ are defined as in Theorem 1.2 and

with

and

Corollary 1.4. Let $k$ be a positive integer and $q = p^k$ with the prime $p\equiv 1(\rm{mod}\ 3)$. Let $\alpha$ be a primitive element of $\mathbb{F}_q$. Denote by $N_3$ (resp. $N_4$) the number of solutions of the equation $x_1^3+x_2^3 = 0$ (resp. $x_1^3+x_2^3+x_3^3 = 0$) over $\mathbb{F}_q$. Then

and

where the integer $u$ is uniquely determined such that

Corollary 1.4 is a special case of ^[17]. If $k = 1$, then Corollary 1.4 is a special case of ^[7].

This paper is organized as follows. In Section 2, we present several basic concepts including the Jacobi sums, and give some preliminary lemmas. In Section 3, we prove Theorems 1.2 and 1.3 and finally, in Section 4, we supply some examples to illustrate the validity of our results.

2.   Preliminary lemmas

In this section, we present some auxiliary lemmas that are needed in the proof of Theorems 1.2 and 1.3.

If $\lambda$ is a multiplicative character of $\mathbb{F}_q$, then $\lambda$ is defined for all nonzero elements of $\mathbb{F}_q$. It is now convenient to extend the definition of $\lambda$ by setting $\lambda(0) = 1$ if $\lambda$ is the trivial character and $\lambda(0) = 0$ otherwise.

For any element $\alpha\in\mathbb{F}_{q} = \mathbb{F}_{p^k}$, the norm of $\alpha$ relative to $\mathbb{F}_p$ is defined by (see, for example, ^[14,15])

For the simplicity, we write $\mathbb{N}(\alpha)$ for $\mathbb{N}_{\mathbb{F}_q/\mathbb{F}_{p}} (\alpha)$. For any $\alpha\in \mathbb{F}_{q}$, it is clear that $\mathbb{N}(\alpha)\in \mathbb{F}_p$. Furthermore, if $\alpha$ is a primitive element of $\mathbb{F}_q$, then $\mathbb{N}(\alpha)$ is a primitive element of $\mathbb{F}_p$.

Let $\chi$ be a multiplicative character of $\mathbb{F}_p$. Then $\chi$ can be lifted to a multiplicative character $\lambda$ of $\mathbb{F}_{q}$ by setting $\lambda(\alpha) = \chi(\mathbb{N}(\alpha))$. The characters of $\mathbb{F}_p$ can be lifted to the characters of $\mathbb{F}_q$, but not all characters of $\mathbb{F}_q$ can be obtained by lifting a character of $\mathbb{F}_p$. The following lemma tells us when $p\equiv 1(\rm{mod}\ 3)$, then any multiplicative character $\lambda$ of order 3 of $\mathbb{F}_q$ can be lifted by a multiplicative character of order 3 of $\mathbb{F}_p$.

Lemma 2.1. ^[15] Let $\mathbb{F}_p$ be a finite field and $\mathbb{F}_q$ be an extension of $\mathbb{F}_p$. A multiplicative character $\lambda$ of $\mathbb{F}_q$ can be lifted by a multiplicative character $\chi$ of $\mathbb{F}_p$ if and only if $\lambda^{p-1}$ is trivial.

Let $\lambda_1, \dots, \lambda_s$ be $s$ multiplicative characters of $\mathbb{F}_q$. The Jacobi sum $J(\lambda_1, \cdots, \lambda_s)$ is defined by

where the summation is taken over all $s$-tuples $(\gamma_1, \cdots, \gamma_s)$ of elements of $\mathbb{F}_q$ with $\gamma_1+\cdots+\gamma_s = 1$. It is clear that if $\sigma$ is a permutation of $\{1, \cdots, s\}$, then

The readers are referred to ^[2] and ^[15] for basic facts on Jacobi sums.

The following theorem is an analog of Hasse-Davenport theorem for Jacobi sums which establishes an important relationship between the Jacobi sums in $\mathbb{F}_q$ and the Jacobi sums in $\mathbb{F}_p$.

Lemma 2.2. ^[15] Let $\chi_1, \dots, \chi_s$ be $s$ multiplicative characters of $\mathbb{F}_p$, not all of which are trivial. Suppose $\chi_1, \dots, \chi_s$ are lifted to characters $\lambda_1, \dots, \lambda_s$, respectively, of the finite extension field $\mathbb{F}_{p^k}$ of $\mathbb{F}_p$. Then

We give the reduction formula for Jacobi sums as follows.

Lemma 2.3. ^[2] Let $\lambda_1, \cdots, \lambda_{s-1}, \lambda_s$ be $s$ nontrivial multiplicative characters of $\mathbb{F}_q$. If $s\geq2$, then

The value of some needed Jacobi sums are listed in the following two lemmas.

Lemma 2.4. ^[2] Let $p\equiv 1(\rm{mod} \ 3)$ be a prime, $q = p^k$, $\alpha$ be a primitive element of $\mathbb{F}_q$, and let $\chi$ be a multiplicative character of order 3 over $\mathbb{F}_p$. Then

where the integers $u$ and $v$ are uniquely determined such that

Lemma 2.5. ^[2] Let $p\equiv 1(\rm{mod} \ 3)$, $g$ be a primitive element of $\mathbb{F}_p$ and let $\chi$ be a multiplicative character of order 3 over $\mathbb{F}_p$ such that $\chi(g) = \frac{-1+i\sqrt{3}}{2}$. Let the integers $u$ and $v$ be defined as in Lemma 2.4. Then the values of the nine Jacobi sums $J(\chi^m, \chi^n)$ $(m, n = 0, 1, 2)$ are given in the following Table 1.

The following lemma gives the number of solutions of the diagonal equation in terms of Jacobi sums.

Lemma 2.6. ^[2] Let $k_1, \dots, k_s$ be positive integers, $a_1, \dots, a_s\in\mathbb{F}_{q}^*$ and let $c\in\mathbb{F}_{q}$. Set $d_i = \gcd(k_i, q-1)$, and let $\lambda_i$ be a multiplicative character of order $d_i$ of $\mathbb{F}_{q}$, $i = 1, \dots, s$. Then the number $N$ of solutions of the equation $a_1x_1^{k_1}+\cdots+a_sx_s^{k_s} = c$ is given by

if $c = 0$, and by

if $c\neq 0$.

3.   Proof of Theorems 1.2 and 1.3

In this section, we give the proof of Theorems 1.2 and 1.3. First, we begin with a lemma.

Lemma 3.1. Let $p\equiv 1(\rm{mod} \ 3)$ be a prime, $q = p^k$, $\alpha$ be a primitive element of $\mathbb{F}_q$ and let $\lambda$ be the multiplicative character of order 3 of $\mathbb{F}_q$ such that $\lambda(\alpha) = \frac{-1+i\sqrt{3}}{2}$. Then for any positive integers $a$, $b$ and $\beta\in\mathbb{F}_q^*$, we have

and

where $E(a, b, k)$ and $O(a, b, k)$ are defined as in Theorem 1.2.

Proof. The first part of the lemma is obvious. Now we focus on the proof of the second part of the lemma. One can divide this into the following three cases.

If ${\rm ind}_{\alpha}\beta\equiv 0 (\rm{mod}\ 3)$, then $\lambda(\beta) = \lambda(\beta^2) = 1$. One has

If ${\rm ind}_{\alpha}\beta\equiv 1 (\rm{mod}\ 3)$, then $\lambda(\beta) = \frac{-1+i\sqrt{3}}{2}$ and $\lambda(\beta^2) = \frac{-1-i\sqrt{3}}{2}$. One has

If ${\rm ind}_{\alpha}\beta\equiv 2 (\rm{mod}\ 3)$, then $\lambda(\beta) = \frac{-1-i\sqrt{3}}{2}$ and $\lambda(\beta^2) = \frac{-1+i\sqrt{3}}{2}$. One has

The result follows immediately from (3.1)–(3.3).

Now we can turn our attention to prove Theorems 1.2 and 1.3.

Proof of Theorem 1.2. Since $\gcd(3, q-1) = 3$, by Lemma 2.6, let $\alpha$ be a primitive element of $\mathbb{F}_q$ and $\lambda$ be the multiplicative character of $\mathbb{F}_q$ of order 3 with $\lambda(\alpha) = \frac{-1+i\sqrt{3}}{2}$. Then we deduce that

if $c = 0$, and

if $c\neq 0$.

Since $p\equiv 1(\rm{mod} \ 3)$, it follows that $\lambda^{p-1}$ is trivial. By Lemma 2.1, the cubic multiplicative character $\lambda$ can be lifted by a cubic multiplicative character $\chi$ of $\mathbb{F}_p$. Combining with the Lemma 2.2, Table 1 of Lemma 2.5 and Lemma 3.1, we obtain

and

Then from (3.4)–(3.7), one can easily deduce the result of Theorem 1.2.

Proof of Theorem 1.3. By the same argument as in the proof of theorem 1.2, let $\alpha$ be a primitive element of $\mathbb{F}_q$ and $\lambda$ be the multiplicative character of $\mathbb{F}_q$ of order 3 with $\lambda(\alpha) = \frac{-1+i\sqrt{3}}{2}$. We deduce that

if $c = 0$, and

if $c\neq 0$.

Similarly, the cubic multiplicative character $\lambda$ can be lifted by a cubic multiplicative character $\chi$ of $\mathbb{F}_p$. By using the Lemmas 2.2, 2.3, Table 1 of Lemma 2.5 and Lemma 3.1, we get

and

By second part of Lemma 3.1, we derive that

Note that Lemma 2.4 tells us $u^2+3v^2 = 4p$. Then from the first part of Lemma 3.1, one has

and

Thus from (3.8)–(3.14), the desired result of Theorem 1.3 follows immediately.

4.   Some examples

In this section, we present some examples to demonstrate the validity of our results.

Example 4.1. Let $q = 13^4$. One can check that 2 is a primitive element of $\mathbb{F}_{13}$. Let $\omega$ be a primitive element of $\mathbb{F}_q$ such that $\mathbb{N}(\omega) = \omega^{\frac{13^4-1}{13-1}} = 2$. We consider the numbers of solutions of the cubic equations

and

over $\mathbb{F}_q$.

Since $\omega^{\frac{13^4-1}{3}} = (\omega^{\frac{13^4-1}{13-1}})^{\frac{13-1}{3}} = 2^4$, the integers $u$ and $v$ in Lemma 2.4 are determined by

We can get that $u = -5$ and $v = -3$. Therefore, by Theorem 1.2, we have

and

Example 4.2. Let $q = 31^2$. One can check that 3 is a primitive element of $\mathbb{F}_{31}$. Let $\omega$ be a primitive element of $\mathbb{F}_{q}$ such that $\mathbb{N}(\omega) = \omega^{\frac{31^2-1}{31-1}} = 3$. We consider the numbers of solutions of the cubic equations

and

over $\mathbb{F}_q$.

Since $\omega^{\frac{31^2-1}{3}} = (\omega^{\frac{31^2-1}{31-1}})^{\frac{31-1}{3}} = 3^{10}$, the integers $u$ and $v$ are determined by

We get $u = 4$ and $v = 6$. Thus by Theorem 1.3, we deduce that

and

Example 4.3. Let $q = 7^3$. It is clear that 3 is a primitive element of $\mathbb{F}_{7}$. Let $\omega$ be a primitive element of $\mathbb{F}_{q}$ such that $\mathbb{N}(\omega) = \omega^{\frac{7^3-1}{7-1}} = 3$. We consider the the numbers of solutions of the cubic equations

and

over $\mathbb{F}_q$.

Similarly, since $\omega^{\frac{7^3-1}{3}} = (\omega^{\frac{7^3-1}{7-1}})^{\frac{7-1}{3}} = 3^2$, the integers $u$ and $v$ are determined by

We deduce that $u = 1$ and $v = -3$. Thus by Theorem 1.3, we have

and

5.   Conlusions

Studying the number of solutions of the polynomial equation $f(x_1, x_2, \cdots, x_n) = 0$ over $\mathbb{F}_{q}$ is one of the main topics in the theory of finite fields. Generally speaking, it is difficult to give an explicit formula for the number of solutions of the equation $f(x_1, x_2, \cdots, x_n) = 0$. There are many researchers who concentrated on finding the formula for the number of solutions of $f(x_1, x_2, \cdots, x_n) = 0$ under certain conditions. Exponential sums are important tools for solving problems involving the number of solutions of the equation $f(x_1, x_2, \cdots, x_n) = 0$ over $\mathbb{F}_{q}$. In this paper, by using the Jacobi sums and the Hasse-Davenport theorem for Jacobi sums, we give an explicit formulae for the numbers of solutions of cubic diagonal equations $a_1x_1^3+a_2x_2^3 = c$ and $b_1x_1^3+b_2x_2^3+b_3x_3^3 = c$ over $\mathbb{F}_q$ are given, with $a_i, b_j\in\mathbb{F}_q^*$ $(1\leq i\leq 2, 1\leq j\leq 3)$, $c\in\mathbb{F}_q$ and $p\equiv1(\rm{mod} \ 3)$. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the cubic diagonal equations $a_1x_1^3+a_2x_2^3+\cdots+a_sx_s^3 = c$ of $s\geq 4$ variables with $a_i\in\mathbb{F}_q^*$ $(1\leq i\leq s)$, $c\in\mathbb{F}_q$ and $p\equiv1(\rm{mod} \ 3)$, can also be deduced.

Acknowledgments

The Authors express their gratitude to the anonymous referee for carefully examining this paper and providing a number of important comments and suggestions. This research was supported by the National Science Foundation of China (No. 12026223 and No. 12026224) and by the National Key Research and Development Program of China (No. 2018YFA0704703).

Conflict of interest

We declare that we have no conflict of interest.