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

1.   Introduction

Let $p$ be an odd prime number with $q = p^s$, $s\in \mathbb Z^+$. Let $\mathbb{F}_q$ be the finite field of $q$ elements. For any polynomial $f(x_1, \cdots, x_n)$ over $\mathbb{F}_q$ with $n$ variables, we let $N(f = 0)$ stand for the number of $\mathbb{F}_q$-rational points on the affine hypersurface $f(x_1, \cdots, x_n) = 0$ over $\mathbb{F}_q^n$. That is, we have

Calculating the value of $N(f = 0)$ is a main topic in finite fields. Weil ^[15] proposed his famous conjecture on the number of rational points of the nonsingular projective hypersurface over $\mathbb{F}_q^n$. However, it is difficult to give an exact formula for $N(f = 0)$. Studying the explicit formula for $N(f = 0)$ under certain conditions has attracted a lot of authors for many years. Some works were done by Ax ^[3], Adolphson and Sperber ^[1,2], Carlitz ^[5], Hong ^[7,8,9], Hu, Hong and Zhao ^[11], Zhao, Hong and Zhu ^[17]. It is noticed that the $p$-adic method is used by Hong et al. in ^[10] to establish the universal Kummer congruences.

On the other hand, in 1977, Chowla, Cowles and Cowles ^[6] determined the number of solutions of the equation

in $\mathbb{F}_p$. In 1981, Myerson ^[13] extend the result in ^[6] to the field $\mathbb{F}_q$ and first studied the number of solution of the equation

over $\mathbb{F}_q$. In 2018, Zhang and Hu ^[16] determined an explicit formula of equation

with $p\equiv 1\pmod 3$ as follows: Let $N(c)$ be the number of solutions of $x_1^3+x_2^3+x_3^3+x_4^3 = c, c\in \mathbb{F}_p^*$ with $p\equiv 1\pmod 3$, and $F_p^{\ast} = \langle g\rangle$. Then they proved the following formula:

In this paper, we investigate the question of counting the number of solutions of the following equation:

with $c\in {\mathbb F}_q^{\ast}$. Actually, we obtain the following result.

Theorem 1.1. Let $F = \mathbb{F}_q$ be the finite field with $q = p^s$ where $p$ is an odd prime and $s\in \mathbb{Z}^+$. Let $c\in \mathbb{F}_q^*$ and $g$ be a primitive element of $\mathbb{F}_q^*$.

$\rm (i)$. If $p\equiv1\pmod8$ or $p\equiv5\pmod8$ and $s$ is even, then

If $p\equiv5\pmod8$ and $s$ is odd, then

where $a+b{\rm i} = (a'+b'{\rm i})^s$ with $a'$ and $b'$ being integers such that

$\rm (ii)$. If $p\equiv 3\pmod 4$ and $q\equiv 1\pmod 4,$ then

where $r$ is uniquely determined by

$\rm (iii)$. If $p\equiv 3\pmod 4$ and $q\equiv 3\pmod 4,$ then

We notice that Theorem 1.1 (ⅰ) for the special case $q = p$ has been mentioned in the book of Jacobsthal's book. Furthermore, Theorem 1.1 (ⅱ) is a special case of Wolfmann ^[14], but we here get it by a different method. From Theorem 1.1, we can easily deduce the following statement.

Corollary 1.1. Let $F = \mathbb{F}_q$ be the finite field with $q = p^s$, where $p\equiv 1 \pmod 4$ is an odd prime and $s\in \mathbb{Z}^+$. Let $c\in \mathbb{F}_q^*$. Then each of the following is true.

$\rm (i)$. If $p\equiv1\pmod4$, then $|N(x_1^4+x_2^4 = c)-q|\le7\sqrt{q}$ for each $q\ge 9$.

$\rm (ii)$. If $p\equiv3\pmod4$ and $q\equiv 1\pmod4$, then $|N(x_1^4+x_2^4 = c)-q|\le(7+4\sqrt{2})\sqrt{q}$ for each $q\ge 81$.

This paper is organized as follows. First of all, in Section 2, we present several basic concepts including the Gauss sums, and give some preliminary lemmas. Then in Section 3, we give the proof of our main result Theorem 1.1 and Corollary 1.2. Finally, in Section 4, we supply two examples.

2.   Preliminaries

In this section, we present several definitions and auxiliary lemmas that are needed in the proof of Theorem 1.1. We begin with three definitions.

Definition 2.1. Let $p$ be a prime number and $q = p^s$ with $s$ being a positive integer. Let $\alpha$ be an element of $\mathbb{F}_{q}$. Then the trace and norm of $\alpha$ relative to $\mathbb{F}_p$ are defined by

and

respectively. For the simplicity, we write $\text{Tr}(\alpha)$ and $\mathbb{N}_(\alpha)$ for $\text{Tr}_{\mathbb{F}_{q}/{\mathbb{F}_p}}(\alpha)$ and $\mathbb{N}_{\mathbb{F}_{q}/{\mathbb{F}_p}}(\alpha)$, respectively.

Definition 2.2. Let $\chi$ be a multiplicative character of $\mathbb{F}_q$ and $\psi$ an additive character of $\mathbb{F}_q$. Then we define the Gauss sum $G(\chi, \psi)$ by

Definition 2.3. Let $\chi _1$ and $\chi _2$ be multiplicative characters of $\mathbb{F}_q$. Then the sum

is called a Jacobi sum in $\mathbb{F}_q$.

The character $\psi_0$ represents the trivial additive character such that $\psi_0(x) = 1$ for all $x\in\mathbb{F}_q$ and $\chi_0$ represents the trivial multiplicative character such that $\chi_0(x) = 1$ for all $x\in\mathbb{F}_q$. For any $x\in \mathbb{F}_q$, let

Then we call $\psi_1$ the canonical additive character of $\mathbb{F}_q$. Let $a\in \mathbb{F}_q$. Then we define

for all $x\in \mathbb{F}_q$. For each character $\psi$ of $\mathbb{F}_q$ there is associated the conjugate character $\overline{\psi}$ defined by $\overline{\psi}(x) = \overline{\psi(x)}$ for all $x\in\mathbb{F}_q$. Let $\eta$ be the quadratic character of $\mathbb{F}_q$.

We give several basic identities about Gauss sums as follows.

Lemma 2.1. ^[12] Each of the following is true:

$\rm(i).$ $G(\chi, \psi_{ab}) = \overline{\chi (a)}G(\chi, \psi_b)$ for $a\in \mathbb{F} _q^*$, $b\in \mathbb{F} _q$.

$\rm(ii).$ $G(\overline\chi, \psi) = \chi (-1)\overline {G(\chi, \psi)}.$

$\rm(iii).$ $\left|G(\chi, \psi)\right| = q^{1/2}$ for $\chi \ne\chi _0$ and $\psi\ne\psi_0.$

Lemma 2.2. ^[12] Let $\mathbb{F}_q$ be a finite field with $q = p^s$, where $p$ is an odd prime and $s\in \mathbb{N}_+$. Then

If $\chi _1$ and $\chi _2$ are nontrivial, there exists an important connection between Jacobi sums and Gauss sums that will allow us to determine the value of Jacobi sums.

Lemma 2.3. ^[12] If $\chi _1$ and $\chi _2$ are multiplicative characters of $\mathbb{F}_q$ and $\psi$ is a nontrivial additive character of $\mathbb{F}_q$, then

if $\chi _1 \chi _2$ is nontrivial.

For a multiplicative character $\chi$ of $\mathbb{F}_q,$ we obviously have $\chi (-1) = \pm 1.$ The value $\chi (-1)$ is of interest. The following result is regarding the sign of $\chi (-1).$

Lemma 2.4. ^[12] Let $\chi$ be a multiplicative character of $\mathbb{F}_q$ of order $n$. Then $\chi (-1) = -1$ if and only if $n$ is even and $\frac{q-1}{n}$ is odd.

Clearly, we have the following consequence.

Corollary 2.1. Let $p \equiv 1 \pmod 4$ be an odd prime and $q = p^s$ with $s$ being a positive integer. Let $\varphi$ be a multiplicative character of $\mathbb{F}_q$ of order $4$. Then

Proof. This corollary follows immediately from Lemma 2.4.

Let $\widehat{\mathbb{F} _q^*}$ be the dual group consisting of all multiplicative characters of $\mathbb{F} _q^*$ with the generator $\varphi$. Then ${\rm ord}(\varphi) = q-1$ and the multiplicative character $\lambda$ with order $d$ with $d|(q-1)$ has the expression $\lambda = \varphi^{\frac{q-1}{d}t^{\prime}}$, where $0\le t^{\prime} < d$ and $\gcd(t^{\prime}, d) = 1$. Furthermore, the number of multiplicative character $\lambda$ with order $d$ is $\phi(d)$, where $\phi$ is Euler's totient function. We also need the following result.

Lemma 2.5. Let $\lambda$ be a multiplicative character of $\mathbb{F}_q$ with order $\gcd(4, q-1)$. Then

Proof. We divide this into the following three cases. Let $\lambda$ be any multiplicative character of $\mathbb{F}_q$ with order $d: = \gcd(4, q-1)$.

CASE 1. $b = 0$. Then $x^4 = 0$ has only zero solution $x = 0$ in $\mathbb{F}_q$. That is, one has $N(x^4 = 0) = 1$. Since ${\lambda}^0(0) = 1$ and ${\lambda}^j(0) = 0$ for $1\le j \le {d-1}$, it follows that

as desired. So part (ⅰ) is proved in this case.

CASE 2. $b\ne 0$ and $x^4 = b$ has a solution in $\mathbb{F}_q$. Let $b = g^k$ and $x = g^y$. Then $x^4 = b$ is equivalent to the congruence

Then the congruence (2.1) has exactly $d = \gcd(4, q-1)$ solutions $y$. Hence $x^4 = b$ has exactly $d$ solutions in $\mathbb{F}_q$. Namely, $N(x^4 = b) = d$.

Let $x_0$ be an element of $\mathbb{F}_q$ with $x_0^4 = b$. For any integer $j$ with $0\le j \le {d-1}$, since $d|4$ implying that $\lambda^4 = \chi_0$, the trivial multiplicative character, we have

Therefore one derives that

as desired. Hence part (ⅰ) holds in this case.

CASE 3. $b\ne 0$ and $x^4 = b$ has no solution in $\mathbb{F}_q$. Then $N(x^4 = b) = 0$ and (2.1) has no solution in $\mathbb{F}_q$. Let $b = g^k$. Then $d\nmid k$ and $\lambda(b) = {\lambda}^k(g)\ne 1$ since $\lambda(g)$ is a $d$-th primitive root of unity. Then

which implies that

Since $\lambda(b)\ne 1$, we have

as required. Part (ⅰ) is proved in this case.

This finishes the proof of Lemma 2.5.

The following relation between the Gauss sum $G(\chi ^\prime, \psi^\prime)$ of $\mathbb{F}_p$ and the Gauss sum $G(\chi, \psi)$ of $\mathbb{F}_q$ is due to Hasse and Davenport.

Lemma 2.6. ^[12] Let $\psi^\prime$ be an additive and $\chi ^\prime$ a multiplicative character of $\mathbb{F}_p$, not both of them trivial. Suppose that $\psi^\prime$ and $\chi ^\prime$ are lifting to characters $\psi$ and $\chi$, respectively, of the finite extension field $\mathbb{F}_q$ of $\mathbb{F}_p$ with $[\mathbb{F}_q:\mathbb{F}_p] = s$. Then

For a certain special multiplicative character of $\mathbb{F}_p,$ the following result gives an explicit formula about the associated Jacobi sums.

Lemma 2.7. ^[4] Let $p\equiv1\pmod4$ be an odd prime number and let $\varphi'$ be a multiplicative character of $\mathbb{F}_p$ with $ord(\varphi') = 4.$ If $\theta$ is a generator of $\mathbb{F}_p^*$ with $\varphi'(\theta) = {\rm i}$, then

where $a'$ and $b'$ are integers such that $a'^2+b'^2 = p$, $a'\equiv-1\pmod4$ and $b'\equiv a'{\theta}^{\frac{p-1}{4}} \pmod p$.

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 result characterizes all the characters of $\mathbb{F}_q$ that can be obtained by lifting a character of $\mathbb{F}_p$.

Lemma 2.8. ^[12] Let $\chi$ be a multiplicative character of $\mathbb{F}_q$ with $q = p^s$. Then $\chi$ can be lifted from a multiplicative character $\chi'$ of $\mathbb{F}_p$ if and only if $\chi ^{p-1}$ is trivial.

3.   Proofs of Theorems 1.1

In this section, we present the proof of Theorem $1.1$ as follows.

Proof of Theorem 1.1. ${\rm(i)}$. Let $p\equiv1\pmod4$. For $x\in\mathbb{F}_q,$ from the trigonometric identity

we can deduce that

Denote

Then by Lemma 2.5, we get

where $\varphi$ is a multiplicative character of $\mathbb{F}_q$ with ${\rm ord}(\varphi) = 4.$ Then $\varphi(g) = \pm{\rm i}$. WLOG, in what follows, we set $\varphi(g) = {\rm i}$.

Note that $\varphi^2 = \eta$ and $\varphi^3 = \overline{\varphi},$ we know that

Since

it follows from the definition of Gauss sum and Lemma 2.1 that

Noticing that the value of $\eta$ is real and $\eta(x) = \Big(\frac{\mathbb{N}(x)}{p}\Big)$, from the Lemmas 2.1 and 2.2 we deduce that

From (3.1)) and ((3.2), we derive that

Since $\varphi^3 = \overline{\varphi}$, $\overline{\varphi^2} = \varphi^2 = \eta$, $\varphi^2(x) = \Big(\frac{\mathbb{N}(x)}{p}\Big)$ and Lemma 2.1 implying that

$G(\varphi, \psi_1)\overline{G(\varphi, \psi_1)} = q$, it follows that

By using the following simple facts

we deduce that

Since $-xc$ runs over $\mathbb{F}_q^*$ as $x$ runs through $\mathbb{F}_q^*$, it follows from (3.3), (3.4) and the fact of $\sum\limits_{x\in\mathbb{F}_q^*}\psi_1(-xc) = -1$ that

Note that

From Lemmas 2.1 and 2.2, we have

Noting that $\varphi(1) = \overline{\varphi}(1) = 1 = \eta(1)$, it follows from (3.5) that

From Lemmas 2.2, 2.3 and 2.6-2.8, we can deduce that

where $a'$ and $b'$ are integers such that

Letting $a+b{\rm i} = (a'+b'{\rm i})^s$ gives us that

Thus

By Corollary 2.1, we get

From (3.5), (3.6), $\varphi(g) = {\rm i}$ and $\eta(g) = -1$ we obtain that

By Corollary 2.1, we have

From (3.5), (3.6), $\varphi(g^2) = -1$ and $\eta(g^2) = 1$, we have

By Corollary 2.1 it follows that

From (3.5), (3.6), $\varphi(g^3) = -{\rm i}$ and $\eta(g^3) = -1$, we deduce

By Corollary 2.1 we have

From (3.7), (3.8), (3.9) and (3.10), we can conclude the proof of part $\rm(i)$.

${\rm(ii)}$. Let $p\equiv3\pmod4$ and $q\equiv1\pmod4.$ Since $p\equiv3\pmod4$ implying that

one must have that $s$ is even. Let $N_k$ be the number of solutions of $x_1^4+x_2^4+\cdots +x_k^4 = 0$ over $\mathbb{F}_q$. Myerson ^[13] gave the value of

where $r$ is uniquely determined by

Since

one has

From (3.11) and Corollary 2.1, we have $\varphi(-1) = 1$ if $s$ is even.

By (3.5), one has

From (3.12) and (3.13), we can deduce

Let $G^2(\varphi, \psi_1) = 3r\sqrt{q}+B{\rm i}.$ By $G^2(\varphi, \psi_1)\overline{G^2(\varphi, \psi_1)} = q^2$, one has

So we can write $G^2(\varphi, \psi_1) = 3r\sqrt{q}\pm\sqrt{q^2-9r^2q}\ {\rm i}.$ From (3.5), one has

This finishes the proof of $\rm(ii)$.

${\rm(iii)}$. Let $p\equiv3\pmod4$ and $q\equiv3\pmod4.$ Since $\gcd(4, q-1) = \gcd(2, q-1).$ By Lemma 2.5, one has $N(x^4 = A) = N(x^2 = A).$ It follows that

This finishes the proof of part $\rm(iii)$, and hence that of Theorem 1.1.

Proof of Corollary 1.2. ${\rm(i)}$. Let $q\equiv1\pmod4$. From Theorem 1.1, one has $|a|\le \sqrt{q}$ and $|b|\le \sqrt{q}$, therefore one deduces that $|a\pm 2b|\le 3\sqrt{q}$. Then we derive the desired result $|N(x_1^4+x_2^4 = c)-q|\le7\sqrt{q}$ by triangle inequality.

${\rm(ii)}$. Let $p\equiv1\pmod4$ and $q\equiv3\pmod4$. From Theorem 1.1, one has $|r|\le \sqrt{q}$, $\sqrt{q-9\sqrt{q}}\le\sqrt{q-9r}\le\sqrt{q+9\sqrt{q}}$ and $|\Big(\big(\pm\overline{\varphi}(-c)\big)+\big(\mp\varphi(-c)\big)\Big){\rm i}|\le 2$. From this, we deduce that

as required. The proof of Corollary 1.2 is complete.

4.   Two Examples

In this final section, we provide two examples to demonstrate the validity of our main result Theorem 1.1.

Example 4.1. For finite field $\mathbb{F}_5$, it is easy to see that $2$ is a generator of $\mathbb{F}_5^*$. Note that $s = 1$. Then $a' = a = -1$ and $b' = b = -2$.

Since ${\rm ind}_2(1)\equiv 0\pmod 4$, ${\rm ind}_2(2)\equiv 1\pmod 4$, ${\rm ind}_2(3)\equiv 3\pmod 4$, ${\rm ind}_2(2)\equiv 2\pmod 4$. From Theorem 1.1, one can compute and get that

Example 4.2. Observe that $x^2-2$ is irreducible over $\mathbb{F}_5$. Let $\alpha$ be a root of $x^2-2$ over its split field. Then $\mathbb{F}_5(\alpha)$ is an extension field of $\mathbb{F}_5$ with order $25$, and we denote it by $\mathbb{F}_{25}$, where

For any ${x_i+y_i}\in \mathbb{F}_{25}$ with $i = 1, 2$, we define

and

By matlab programme, we confirm that the element $2+4 \alpha$ is a generator of $\mathbb{F}_{25}$.

Note that $s = 2$, $a' = -1$, $b' = -2$. Then $a = -3$ and $b = 4$. Since ${\rm ind}_{2+4\alpha}(1)\equiv 0\pmod 4$ and ${\rm ind}_{2+4\alpha}(\alpha)\equiv 3\pmod 4$, it follows from Theorem 1.1 that

Acknowledgments

The authors are thankful for the anonymous referees for their careful reading of the manuscript and helpful comments. They also thank Dr. Chaoxi Zhu and Dr. Lingfeng Ao for many helpful suggestions.

Conflict of interest

We declare that we have no conflict of interest.