A note on some diagonal cubic equations over finite fields

1.   Introduction

Let $\mathbb{F}_q$ be a finite field of

elements. Let $\mathbb{F}^{\ast}_q$ be the multiplicative group of $\mathbb{F}_q$, i.s.,

Counting the number of solutions $(x_1, x_2, \cdots, x_n)\in \mathbb{F}^n_q$ of the general diagonal equation

over $\mathbb{F}_q$ is an important and fundamental problem in number theory and finite field. The special case where all the $d_i$ are equal has extensively been studied by many authors (see, for example, ^[1,2,3] for $d_i = 3$, and ^[4,5] for $d_i = 4$).

For a prime

Chowla et al. ^[6,7,8] first considered the number of solutions of equation

over finite field $\mathbb{F}_p$.

For $z$ is cubic, using the classic result of the cubic equation of periods by Gauss ^[9]. Chowla et al. ^[10] showed that the number of solutions of (1.1) is $p^2+c(p-1)$, where $c$ is uniquely determined by

For $z$ is non-cubic, as pointed out in ^[6], using the classic result of the cubic equation of periods by Gauss ^[9], one can only obtain that the number of solutions of (1.1) is

where $c$ is uniquely determined, and $d$ is determined except for the sign by

Thus the key of these problems is to determine the sign of $d$. Chowla et al. ^[6] determined the sign of $d$ for the case of 2 being non-cubic in $\mathbb{F}_p$.

Theorem 1.1. ^[6] Let a prime be

If 2 is non-cubic in $\mathbb{F}_p$, then for any non-cubic element $z$, the number of solutions of (1.1) is

where $c$ and $d$ are uniquely determined by (1.2) with

where $H$ is the subgroup of nonzero cubes in $\mathbb{F}^{\ast}_p$.

Therefore, it is nature to ask the following problem: Is there another element which can determine the sign of $d$?

In this paper, we extend the result of Chowla et al. to finite field $\mathbb{F}_q$, and determine the sign of $d$ by non-cube 3.

In the rest of this paper, $\mathbb{F}_q$ is a finite field of

elements with

and $\mathbb{F}^{\ast}_q$ is the multiplicative group of $\mathbb{F}_q$, i.s.,

For any $z\in \mathbb{F}_q$, one lets $A_n(z)$ denote the number of solutions of the following diagonal equation

over $\mathbb{F}_q$. Let $B_n(z)$ be the number of solutions of diagonal cubic equation

over $\mathbb{F}_q$. Since

the nonzero cubic elements form a multiplicative subgroup $H$ of order $\frac{1}{3}(q-1)$ and index 3, which partitions $\mathbb{F}^{\ast}_q$ into three cosets $H, zH$ and $z^2H$. Now, we can state our main results.

Theorem 1.2. Let $\mathbb{F}_q$ be a finite field of

elements with the prime

$c$ is uniquely determined, and $d$ is determined except for the sign by

(1) If $2\mid d$, then 2 is a cube in $\mathbb{F}_q$ and one has

(2) If $2\nmid d$, then 2 is a non-cube in $\mathbb{F}_q$, and for any non-cubic $z$, one has

and

where $d$ is uniquely determined by (1.3) and

Theorem 1.3. Let $\mathbb{F}_q$ be a finite field of

elements with the prime

$c$ is uniquely determined, and $d$ is determined except for the sign by (1.3).

(1) If $3\mid d$, then 3 is a cube in $\mathbb{F}_q$, and one has

(2) If $3\nmid d$, then 3 is a non-cube in $\mathbb{F}_q$, and for any non-cubic $z$, one has

and

where $d$ is uniquely determined by (1.3) and $d\equiv -1 (\text{mod}3)$.

Remark 1.4. (1) When $z$ is cubic in $\mathbb{F}_q$ with

as pointed out in ^[11] (or ^[12]), one has

(2) When

it is known that every element is a cube. When

with

as [13, Theorem 16], one has

and $d = 0$.

(3) In ^[12], Hong and Zhu use the generator $g$ of group $\mathbb{F}^{\ast}_q$ to determine the sign of $d$, and give the sign of $d$ by

where $r_1$ and $r_2$ are uniquely determined by

Here, the norm $\mathrm{N}_{\mathbb{F}_q/\mathbb{F}_p} (\alpha)$ of $\alpha\in \mathbb{F}_q$ over $\mathbb{F}_p$ is defined by

Recently, the authors ^[14] determined the sign of $d$ by

In this paper, we give a more effective way to determine the sign of $d$ for the cases of 2 or 3 being non-cubic.

Using the author's methods in ^[14] and Theorems 1.2 and 1.3, we immediately obtain the following corollaries:

Corollary 1.5. Let $\mathbb{F}_q$ be a finite field of

elements with the prime

$c$ is uniquely determined, and $d$ is determined except for the sign by (1.3). If $2\nmid d$, then for any non-cubic element $z$, one has

and

where $d$ is uniquely determined by (1.3) and

Corollary 1.6. Let $\mathbb{F}_q$ be a finite field of

elements with the prime

$c$ is uniquely determined, and $d$ is determined except for the sign by (1.3). If $3\nmid d$, then for any non-cubic element $z$, one has

and

where $d$ is uniquely determined by (1.3) and

2.   Auxiliary lemmas

Lemma 2.1. ^[15] Let $\mathbb{F}_q$ be a finite field. Let $\psi$ be a nontrivial additive character of $\mathbb{F}_q$. Then, for any $a\in \mathbb{F}_q$, we have

For any $a\in \mathbb{F}^{\ast}_q$, we defined the Gauss sums

Lemma 2.2. ^[13] Let $\mathbb{F}_q$ be the finite field of $q = p^k$ elements with the prime

and $z$ is non-cubic in $\mathbb{F}^{\ast}_q$. Then, $S_1$, $S_z$, and $S_{z^2}$ are the roots of the cubic equation

where $c$ is uniquely determined by

Lemma 2.3. With the conditions of Lemma 2.2, one has

Proof. Since

the nonzero cubic elements form a multiplicative subgroup $H$ of order $\frac{1}{3}(q-1)$ and index 3, which partitions $\mathbb{F}^{\ast}_q$ into three cosets $H$, $zH$, and $z^2H$. Then, for any $a\in z^jH$, we have

For any $b\in \mathbb{F}^{\ast}_q$, we have

Thus, we have

Then, by Lemma 2.1, we have

since $-1$ is cubic in $\mathbb{F}^{\ast}_q$. □

Lemma 2.4. With the conditions of Lemma 2.2, one has

where $c$ is uniquely determined by (2.1).

Proof. By lemma 2.2, we have

and

Then, by Lemma 2.3, we have

□

Lemma 2.5. With the conditions of Lemma 2.2, one has $A_2(z)$ as one of the values

and $A_2(z^2)$ is the other, where $c$ is uniquely determined, and $d$ is determined except for the sign by (2.1).

Proof. Similar to the proof of Lemma 2.4, we have

By Lemma 2.2, we have

and

Then Lemma 2.5 follows from (2.2)–(2.5).

□

Lemma 2.6. We have

Proof. By the definition of $B_2(z)$ and $A_2(z)$, we have

□

Lemma 2.7. Let $\mathbb{F}_q$ be the finite field of

elements with the prime

$c$ is uniquely determined, and $d$ is determined except for the sign by (1.3). Then, 2 is a cube in $\mathbb{F}^{\ast}_q$ if, and only if, $2\mid d$; 3 is a cube in $\mathbb{F}^{\ast}_q$ if, and only if, $3\mid d$.

Proof. Since $\mathbb{F}^{\ast}_q$ is a cyclic group, 2 is cubic in $\mathbb{F}^{\ast}_q$ if, and only if,

By the Euler theorem, we have

Note that

then we have

By [16, Theorem 7.1.1], 2 is cubic in $\mathbb{F}_p$ if, and only if, $2\mid d_0$, where $c_0$ and $d_0$ are determined by

That is,

if, and only if, $2\mid d_0$. Thus, we have that 2 is cubic in $\mathbb{F}^{\ast}_q$ if, and only if, $2\mid d_0$ or $3\mid k$.

Next, we will prove that $2\mid d_0$ or $3\mid k$ if, and only if, $2\mid d$. Since

we have $2\mid c$ if, and only if, $2\mid d$. As pointed out in [14, Lemma 2.8], in integeral ring $\mathcal{O}_K$ of cubic cyclotomic field

we have

So, we have

If $2\mid d_0$, it is easy to see that $2\mid c$. If $2\nmid d_0$, then $2\mid \frac{c_0+3d_0}{2}$ or $2\mid \frac{c_0-3d_0}{2}$, since

When $2\mid \frac{c_0+3d_0}{2}$, we have

When $2\mid \frac{c_0-3d_0}{2}$, we have

Thus we have

Hence, we have 2 is cubic in $\mathbb{F}^{\ast}_q$ if, and only if, $2\mid d_0$, or $3\mid k$ if, and only if, $2\mid c$, or if and only if $2\mid d$. Similarly, we can also prove that 3 is a cube in $\mathbb{F}^{\ast}_q$ if, and only if, $3\mid d$. □

3.   Proof of Theorem 1.2

Let $\mathbb{F}_q$ be a finite field of

elements with

Then the nonzero cubic elements form a multiplicative subgroup $H$ of order $\frac{1}{3}(q-1)$. We let

Since for any $a\in H$, the equation $x^3 = a$ in $\mathbb{F}_q$ exactly has three different solutions, if 2 is non-cubic in $\mathbb{F}^{\ast}_q$, then it is easy to see that

Lemma 3.1. If 2 and $z$ are non-cubic in $\mathbb{F}^{\ast}_q$, then

Proof. Since 2 and $z$ are non-cubic in $\mathbb{F}^{\ast}_q$, $z\in 2H\cup 4H$. If $z\in 2H$, i.e.,

there is a $h\in \mathbb{F}^{\ast}_q$ and $z = 2h^3$. Then, it is easy to see that

Similarly, if

we have

□

Lemma 3.2. We have $M\equiv 1(\text{mod}2)$.

Proof. Let

Obviously, if $(a, b)\in \mathcal{M}_1$, then $(b, a)\in \mathcal{M}_1$. Thus, we have $M_1$ is even. If $(a, a)\in \mathcal{M}$, then $a = 1$. Hence, $M$ is odd. □

Proof of Theorem 1.2. If $2\mid d$, then 2 is a cube in $\mathbb{F}_q$ by Lemma 2.7, and one has

as pointed out in ^[11] (or ^[12]).

If $2\nmid d$, then 2 is non-cubic in $\mathbb{F}^{\ast}_q$ by Lemma 2.7. Then, by Lemma 2.5, we can assume that

Next, we begin to determine the sign of $d$. By (3.1) and Lemma 3.2, we have

Since

and a prime

we have

So, we have

and then

Thus, by Lemma 2.5, we have

Finally, Theorem 1.2 immediately follows from Lemmas 2.6 and 3.1. □

4.   Proof of Theorems 1.3 and an example

Let

Then, if 3 is non-cubic, it is easy to see that

Lemma 4.1. If 3 and $z$ are non-cubic in $\mathbb{F}^{\ast}_q$, then

Proof. This is similar to the proof of Lemma 3.1. □

Lemma 4.2. If 3 is non-cubic, we have

Proof. We divide the set $\mathcal{N}(z)$ into three disjoint subsets,

where

Let

Then, it is easy to see that

Since

and a prime

we have

Thus we have

Since 3 is non-cubic, then

So, we have

□

Proof of Theorem 1.3. If $3\mid d$, then 3 is a cube in $\mathbb{F}_q$ by Lemma 2.7, and one has

as pointed out in ^[11] (or ^[12]).

If $3\nmid d$, then 3 is non-cubic in $\mathbb{F}^{\ast}_q$ by Lemma 2.7. Then, by Lemma 2.5, we can assume that

then

Next, we begin to determine the sign of $d$.

By (4.1), we have

Then, by Lemma 2.4, we have

and by Lemma 4.2, we have

Thus, by Lemma 2.5, we have

with

Hence, Theorem 1.3 immediately follows from Lemmas 2.6 and 4.1. □

Example 4.3. We take

If the integers $c$ and $d$ satisfy that

then

Thus, 2 is cubic and 3 is non-cubic. Then, it follows from Theorem 1.3 that the numbers $A_2(3)$ and $B_2(3)$ of the cubic equations

are given by

and

respectively.

5.   Conclusions

In this paper, we study the number of solutions of equations:

and

in finite field $\mathbb{F}_q$ with

When

for any $z$ is non-cubic in $\mathbb{F}_p$. In 1978, Chowla et al. determined the sign of $d$ for the case of $z = 2$ being non-cubic in $\mathbb{F}_p$. In Theorem 1.2, we extend the result of Chowla et al. to finite field $\mathbb{F}_q$. In Theorem 1.3, we establish a new method to determine the sign of $d$ for the case of $z = 3$ being non-cubic. Moreover, we think it is interesting to find a more effective way to determine the sign of $d$ for the case of $z > 3$.

Author contributions

Wenxu Ge: writing-review and editing, writing-original draft, validation, resources, methodology, formal analysis, conceptualization. Weiping Li: writing-review and editing, resources, methodology, supervision, validation, formal analysis. Tianze Wang: resources, methodology, supervision, validation, formal analysis, funding acquisition. All authors read and approved the final manuscript.

Use of AI tools declaration

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

Acknowledgments

The authors are partially supported by the National Natural Science Foundation of China (Grant Nos. 11871193 and 12071132), the Natural Science Foundation of Henan Province (Nos. 222300420493 and 232300421223) and the China Scholarship Council (No. 202308410506).

Conflict of interest

The authors declare that they have no conflicts of interest.