On the number of solutions of two-variable diagonal sextic equations over finite fields

1.   Introduction

Let $p$ be a prime, $k$ a positive integer, $q = p^k$, and let $\mathbb{F}_q$ be the finite field of $q$ elements. Let $f$ be a polynomial over $\mathbb{F}_q$ with $n$ variables, and denote by

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

Studying the value of $N(f, q)$ is one of the main topics in the theory of finite fields. Generally speaking, it is difficult to give the explicit formula for $N(f, q)$.

The degree $d$ of the polynomial $f$ plays an important role in the estimate of $N(f, q)$. An upper bound for $N(f, q)$ ^[14] is given by

For any positive integer $m$, we use ${\rm{ord}}_pm$ to denote the $p$-adic valuation of $m$. Suppose that $N(f, q) > 0$, the classical Chevalley-Warning theorem shows that ${\rm{ord}}_pN(f, q) > 0$ if $n > d$. Furthermore, let $\lceil x\rceil$ denote the least integer $\geq x$ and let $q = p^k$, Ax ^[1] showed that

Finding the explicit formula for $N(f, q)$ under certain conditions has attracted researchers for many years. From ^[13,14] we know that there exists an explicit formula for $N(f, q)$ satisfying ${\rm{deg}}(f) \leq2$ over $\mathbb{F}_{q}$. Some other works were done by Baoulina ^[2,3,4,5], Cao et al. ^[7,8], Hua and Vandiver ^[12], Hu et al. ^[10,11], Weil ^[17], and Zhang and Wan ^[20,21].

In 1977, Chowla, Cowles and Cowles ^[9] got a formula for the number of solutions of the hypersurface

over $\mathbb{F}_{p}$. In 1979, Myerson ^[15] extended the result in ^[9] to the field $\mathbb{F}_{q}$ and studied the number of solutions of the equation

over $\mathbb{F}_{q}$. For $q = p^{2t}$ with $p^r\equiv -1({{\rm{mod}}} \ d)$ for a divisor $r$ of $t$ and $d\mid (q-1)$, Wolfmann ^[18] gave an explicit formula of the number of solutions of the equation

over $\mathbb{F}_{q}$ in 1992, where $a_1, a_2, \dots, a_n\in \mathbb{F}_{q}^*$ and $c\in \mathbb{F}_{q}$. In 2018, Zhang and Hu ^[19] determined an explicit formula of the number of solutions of the equation

over $\mathbb{F}_{p}$, with $p\equiv1({\rm{mod}} \ 3)$ and $c\in\mathbb{F}_p^*$. In 2020, J. Zhao et al. ^[22,23] investigated the number of solutions of the forms

and

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

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

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

It is well-known that (see ^[13], p. 105)

over $\mathbb{F}_q$. Let $q = p^k$ with $p$ a prime. If $p = 2$, then $\gcd(6, q-1) = 1$ or $3$ depends on $k$ is odd or even. From Corollary 4 in ^[18], we can obtain the following result.

Theorem 1.1. Let $p = 2$, $k$ an integer, $q = p^k$ and $c\in\mathbb{F}_q^*$. Then

if $k$ is odd, and

if $k$ is even and $n\geq2$, with $\xi = (-1)^{\frac{k}{2}+1}$.

If $p = 3$ and $k$ is an integer, or $p\equiv5({\rm{mod}} \ 6)$ and $k$ is an odd integer, then $\gcd(6, q-1) = 2$. It follows from Theorems 6.26 and 6.27 in ^[14] that the following result is given.

Theorem 1.2. Let $p = 3$ and $k$ an integer, or $p\equiv5({\rm{mod}} \ 6)$ be a prime and $k$ an odd integer, $q = p^k$ and $c\in\mathbb{F}_q^*$. Then

if $n$ is even, and

if $n$ is odd, where $\eta$ is the quadratic multiplicative character of $\mathbb{F}_q$.

If $p\equiv5({\rm{mod}} \ 6)$ and $k$ is an even integer, Hua and Vandiver ^[12] studied the number of solutions of some trinomial equations over $\mathbb{F}_q$ and Wolfmann ^[18] also got the number of solutions of certain diagonal equations over $\mathbb{F}_q$. The following result can be deduced from Corollary 4 of ^[18].

Theorem 1.3. Let $p\equiv5({\rm{mod}} \ 6)$ be a prime, $k$ an even integer, $q = p^k$, $n\ge 2$ and $c\in\mathbb{F}_q^*$. Then

where $\xi = (-1)^{\frac{k}{2}+1}$.

However, the formula for $N(x_1^6+x_2^6+\cdots+x_n^6 = c)$ is still unknown when $p\equiv1({\rm{mod}} \ 6)$. 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 case with 2 variables. The case with arbitrary $n\ge 3$ variables can be deduced from the reduction formula for Jacobi sums.

Let $g \in\mathbb{F}^*_q$ 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 $\beta = g^r$. Such $r$ is called the index of $\beta$ with the primitive element $g$, and denoted by ${\rm{ind}}_g\beta: = r$.

For any element $\alpha\in E = \mathbb{F}_{p^k}$ and $F = \mathbb{F}_p$, the norm of $\alpha$ relative to $\mathbb{F}_p$ are defined by (see, for example, ^[6,14])

For the simplicity, we write $\mathbb{N}(\alpha)$ for $\mathbb{N}_{E/F} (\alpha)$.

The main result of this paper is stated as follows.

Theorem 1.4. Let $t$ and $k$ be positive integers, $p = 6t+1$ be a prime, $q = p^k$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $c\in\mathbb{F}_q^*$, $g$ be a primitive element of $\mathbb{F}_q$ and $Z = ind_{\mathbb{N}(g)}2$. Then

when $t$ or $k$ is even, and

when both $t$ and $k$ are odd, where $a+ib\sqrt{3} = (a'+ib'\sqrt{3})^k$, $u+iv\sqrt{3} = (u'+iv'\sqrt{3})^k$, $r+is\sqrt{3} = (r'+is'\sqrt{3})^k$ with $a'$ and $b'$ being integers such that

and the integers $r'$, $s'$, $u'$ and $v'$ are given by

This paper is organized as follows. In Section 2, we recall some useful known results which will be needed later. In Section 3, we prove Theorem 1.4 and then present an example to illustrate the validity of our result.

2.   Preliminary lemmas

In this section, we present some useful lemmas that are needed in the proof of Theorem 1.4. For any multiplicative character $\lambda$ 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.

Let $\lambda_1, \cdots, \lambda_s$ 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

Also for any $\alpha\in \mathbb{F}_q^*$, we have that

The readers are referred to ^[6] and ^[14] 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 2.1. ^[14] Let $\chi_1, \cdots, \chi_s$ be $s$ multiplicative characters of $\mathbb{F}_p$, not all of which are trivial. Suppose $\chi_1, \cdots, \chi_s$ are lifted to characters $\lambda_1, \cdots, \lambda_s$, respectively, of the finite extension field $E$ of $\mathbb{F}_p$ with $[E:\mathbb{F}_p] = k$. Then

Let $\chi$ be a multiplicative character of $F$ and $\lambda$ be a multiplicative character of $E$. Recall that $\chi$ can be lifted to $E$ 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 the characters of $\mathbb{F}_q$ can be obtained by lifting a character of $\mathbb{F}_p$. The following lemma tells us when $p\equiv 1({\rm{mod}}\ 6)$, then the multiplicative character $\lambda$ of order 6 of $\mathbb{F}_q$ can be lifted by a multiplicative character of order 6 of $\mathbb{F}_p$.

Lemma 2.2. ^[14] 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 $g$ be a generator of $\mathbb{F}_q^* = \mathbb{F}_{p^k}^*$. Since

and

one knows that $\mathbb{N}(g)$ is a generator of $\mathbb{F}_p^*$. Then we can state the following lemma.

Lemma 2.3. ^[6] Let $p\equiv 1({\rm{mod}} \ 6)$ be a prime, $q$ a power of $p$, $g$ be a generator of $\mathbb{F}_q^*$, and let $\chi$ be a multiplicative character of order 6 over $\mathbb{F}_p$. Then

where the integers $a'$ and $b'$ are uniquely determined by

Lemma 2.4. ^[6] Let $p = 6t+1$ be a prime number. Let $g'$ be the generator of $\mathbb{F}_p^*$ and $\chi$ be a multiplicative character of order $6$ over $\mathbb{F}_p$ such that $\chi(g') = \frac{1+i\sqrt{3}}{2}$. Let the integers $a'$ and $b'$ be defined as in Lemma 2.3 and the integers $u'$, $v'$, $r'$, $s'$ are given as in Theorem 1.4. The values of the 36 Jacobi sums $J(\chi^m, \chi^n)$ $(m, n = 0, 1, 2, 3, 4, 5)$ are given in the following Table 1.

The following lemma gives an explicit formula for the number of solutions of the diagonal equation in terms of Jacobi sums.

Lemma 2.5. ^[6,12,17] Let $k_1, \cdots, k_s$ be positive integers, $a_1, \cdots, a_s, c\in\mathbb{F}_{q}^*$. Set $d_i = \gcd(k_i, q-1)$, and let $\lambda_i$ be a multiplicative character on $\mathbb{F}_{q}$ of order $d_i$, $i = 1, \cdots, s$. Then the number $N$ of solutions of the equation $a_1x_1^{k_1}+\cdots+a_sx_s^{k_s} = c$ is given by

3.   Proof of Theorem 1.4

In this section, we give the proof of Theorem 1.4.

Proof. Let $g$ be a primitive element of $\mathbb{F}_q$ and $\lambda$ be a multiplicative character of $\mathbb{F}_q$ of order 6 with $\lambda(g) = \frac{1+i\sqrt{3}}{2}$. Since $q\equiv 1({\rm{mod}} \ 6)$, then $\gcd(6, q-1) = 6$. Using Lemma 2.5, we have

Since $p\equiv 1({\rm{mod}} \ 6)$, it follows that $\lambda^{p-1}$ is trivial. By Lemma 2.2, the multiplicative character $\lambda$ can be lifted by a multiplicative character $\chi$ of order 6 of $\mathbb{F}_p$. By Lemma 2.1, we obtain

Consider the case when $t$ or $k$ is even. Noting that $\lambda(1) = 1$, from (3.1), one has

From Table 1 of Lemma 2.4, we derive that

and

Thus if ${\rm{ind}}_gc\equiv 0 ({\rm{mod}}\ 6)$, then it follows from (3.2)–(3.4) that

From (3.1), we have

Noting that $\lambda(g) = \frac{1+i\sqrt{3}}{2}$ and $\lambda^6(g) = 1$, then from Table 1 of Lemma 2.4, we deduce that

and

Thus if ${\rm{ind}}_gc\equiv 1 ({\rm{mod}}\ 6)$, from (3.5)–(3.7), one can deduce that

In the similar way, one can deduce that

The case when both $t$ and $k$ are odd can also be proved by the same argument.

Remark. (Reduction formula for Jacobi sums) ^[6] Let $\chi_1, \cdots, \chi_k$ be $k$ nontrivial multiplicative characters of $\mathbb{F}_q$. If $k\geq2$, then

One can use the reduction formula for Jacobi sums to give an explicit formula for the number of solutions of the diagonal sextic equation

of $n\geq3$ variables over $\mathbb{F}_{p^k}$, with $c\in\mathbb{F}_{p^k}^*$ and $p\equiv1({\rm{mod}} \ 6)$. But we omit the tedious details here.

By concluding this section, we present an example to demonstrate the validity of Theorem 1.4.

Example. Let $q = 19^3$. We consider the numbers of solutions of the sextic equation

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

Since 2 is a primitive root modulo 19, we have $Z = {\rm{ind}}_22 = 1$. Let $g$ be a generator of $\mathbb{F}_{19^3}^*$ such that

That means

The integers $a'$ and $b'$ are determined by

We can get that $a' = -4$, $b' = -1$. Since $Z\equiv 1({{\rm{mod}} \ 3})$, we obtain that $r' = 7$, $s' = -3$, $u' = 1$ and $v' = 5$. Thus we have $a = -28$, $b = -45$, $r = -224$, $s = -360$, $u = -224$ and $v = -360$. By Theorem 1.4, one can get that

4.   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 formula for the number of solutions of the two-variable diagonal sextic equations $x_1^6+x_2^6 = c$ over $\mathbb{F}_q$, with $c\in\mathbb{F}_q^*$ and $p\equiv1({\rm{mod}} \ 6)$, where $p$ is the characteristic of $\mathbb{F}_q$. Furthermore, by using the reduction formula for Jacobi sums, the number of solutions of the diagonal sextic equations $x_1^6+x_2^6+\cdots+x_n^6 = c$ of $n\geq3$ variables with $c\in\mathbb{F}_q^*$ and $p\equiv1({\rm{mod}} \ 6)$, 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.