An improvement on Weil bounds for character sums of polynomials over finite fields

Abstract

Let $\mathbb {F}_q$ be a finite field with q elements, where q is a power of a prime p. In this paper, we obtain an improvement on Weil bounds for character sums associated to a polynomial f(x) over $\mathbb {F}_q$, which extends the results of Wan et al. (Des. Codes Cryptogr. 81, 459–468, 2016) and Wu et al. (Des. Codes Cryptogr. 90, 2813–2821, 2022).

1 Introduction

Let $\mathbb {F}_q$ be a finite field with q elements, where $q=p^m$, p is a prime and m is a positive integer. The trace function from $\mathbb {F}_{q}$ onto $\mathbb {F}_p$ is defined by ${\text {Tr}}_{q/p}(x)=x+x^p+\cdots +x^{p^{m-1}}$, $x\in \mathbb {F}_{q}$. The canonical additive character of $\mathbb {F}_{q}$ is defined as follows:

$$\psi _1: \mathbb {F}_{q}\longrightarrow \mathbb {C}^{*}, \psi _1(x)=\zeta _{p}^{{\text {Tr}}_{q/p}(x)},$$

where $\zeta _{p}=e^{\frac{2\pi \sqrt{-1}}{p}}$ denotes the p-th primitive root of unity. It is well-known that an arbitrary nontrivial additive character $\psi$ of $\mathbb {F}_q$ can be expressed by $\psi =\psi _1(c), c\in \mathbb {F}_q^*$.

Let f(x) be a polynomial over $\mathbb {F}_q$ of degree n with $\gcd (n, p)=1$. Define the character sum $S( f, \psi )=\sum _{x\in \mathbb {F}_q}\psi (f(x))$. A well-known result for $S( f, \psi )$ says that

$$\begin{aligned} \bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x)) \bigg |\le (n-1)\sqrt{q}. \end{aligned}$$

(1.1)

The upper bound in (1.1) is called the Weil bound. In general, the bound is tight and cannot be improved except some special families of conditions.

For a positive integer $d\le q= p^m$, the p-ary expansion of $d=d_0+d_1p+\ldots +d_{m-1}p^{m-1}$, the p-ary weight of d, denoted by $W_p(d)$, is equal to $\sum _{i=0}^{m-1}d_i$. In [6], let $f(x)=ax^d+g(x)$ be a polynomial defined over $\mathbb {F}_{q }$, where d is the only exponent in the support of f(x) with p-ary weight equal to $\deg _p(f)$ and $\deg _p(f)=\max \limits _{d\in supp(f)}\{W_p(d)\}$, Gillot obtained the Weil bound of f(x) under the assumption $d = 1 + d_rp^r$ with $\gcd (p, d_r)= 1$ is

$$ |\sum _{x\in \mathbb {F}_{q}}\psi (f(x)) |\le (W_p(d)-1)^m\sqrt{q}. $$

Later, Gillot and Langevin [7] gave a further improvement on this bound.

In [5], let $q=2^m$, for some positive integer N, suppose that $f(x)=x^c+\sum _{i=1}^N\beta ^ix^{ a_i(2t-1)}$, where $\beta _i\in \mathbb {F}_{q}$, t|m, $\gcd (c,q-1)=1$, and each $a_i$ is a positive integer, Gangopadhyay obtained that

$$ |\sum _{x\in \mathbb {F}_{2^m}}\psi (f(x)) |\le 2^{m-t+1}, $$

which improved the Weil bound when $\deg (f(x))\ge 2^{\frac{m}{2} -t+1}+1$.

In [15], suppose that $f(x)=x^n$, where $p^{\frac{2}{m}}\le n\le p^{\frac{m}{2}+\frac{1}{2}}$, Shparlinski estimated the character sum $\bigg |\sum _{x\in \mathbb {F}_q}\psi (x^n) \bigg |$; later, Bourgain and Chang in [2] also obtained a nontrivial estimate of $\bigg |\sum _{x\in \mathbb {F}_q}\psi (x^n)\bigg |$ under the assumption $\gcd (n, \frac{q-1}{p^v-1} )<p^{-v} q^{1-\varepsilon }$, where $q=p^m$, $1\le v< m$,v|m, and $\varepsilon >0$, which extended the results of Bourgain et al. in [3].

Furthermore, there are a lot of attentions on improving the Weil bound for the Artin-Schreier curves, that is $y^q-y=f(x)$ for $(x,y)\in \mathbb {F}_{q^m}^2$, because of their applications in coding theory and computer sciences, which are referred as [4, 9, 13, 14] and therein. About character sums for polynomials, there are a lot of papers to investigate them and presented their applications in cryptography radar and wireless communication systems. Wu et al. in [22] computed the cross correlation distributions of a p-ary m-sequence for several decimations with $d=\frac{q-1}{p^m}$ in case of $p=2$ and $p=3$. Li and Wu et al. in [10, 12, 20, 21] gave some low valued Walsh spectrums of monomial functions. Heng et al. in [8, 11] constructed two families of linear codes with a few weights based on special polynomials over finite fields and some near MDS codes which are optimal locally recoverable codes.

In [18], for character sums associated to a polynomial f(x) over $\mathbb {F}_q$, Wan and Wang presented a new bound, which is called an index bound. They improved the Weil bound for high degree polynomials with small indices. The idea was to use monomial bounds over each cyclotomic cosets because each polynomial can be presented as a cyclotomic mapping. Recently, the index bound was slightly improved by Wu et al. in [19] by using the least index obtained from all possible conjugated exponents of the polynomial f(x).

In this paper, let $\mathbb {F}_q$ be a finite field, where $q=p^m$ and p is a prime. Let $\psi$ be an arbitrary nontrivial additive character of $\mathbb {F}_q$. For an arbitrary polynomial $f(x)\in \mathbb {F}_q[x]$, we investigate the Weil bound of character sum $\sum _{x\in \mathbb {F}_q}\psi (f(x))$ associated to f(x) by considering all possible conjugated exponents of f(x) and present an improved upper bound compared to the results of Wan et al. in [18] and Wu et al. in [19].

2 An improvement of Weil bound for character sums of polynomials

In this section, we always assume that $\mathbb {F}_q$ is a finite field, where $q=p^m$, p is a prime, and m is a positive integer. Let $\gamma$ be a primitive element of $\mathbb {F}_q$, i.e., $\mathbb {F}_{q}^*=\langle \gamma \rangle$ and l, s two positive integers with $ls=q-1$, then we define the cyclotomic cosets of order l in $\mathbb {F}_q$ as follows:

$$C_i=\gamma ^i\langle \gamma ^l\rangle , i = 0, 1,\ldots , l-1,$$

where $C_0=\langle \gamma ^l\rangle$ is a cyclic subgroup of $\mathbb {F}_{q}^*$ generated by $\gamma ^l$.

For any $a_0, a_1, \cdots , a_{l-1} \in \mathbb {F}_q$ and a positive integer r, Wang [16] defined the r-th order cyclotomic mapping $f_{a_0, a_1, \cdots , a_{l-1}}^r$ of index l from $\mathbb {F}_q$ to itself:

$$\begin{aligned}f_{a_0, a_1, \cdots , a_{l-1}}^r=\left\{ \begin{array}{ll} 0 &{} \text{ if } {x= 1}, \\ a_ix^r &{} \text{ if } {x \in C_{i},i=0,1,\cdots ,l-1}. \end{array}\right. \end{aligned}$$

It is shown that the r-th order cyclotomic mappings of index l produce the polynomials of the form $f(x)=x^{r}h( x^{\frac{q-1}{l}})$. More generally any nonconstant polynomial f(x) with degree $\le q-1$ over $\mathbb {F}_q$ such that $f(0)=a_0$ can be written uniquely as $f(x)=x^{r}h( x^{\frac{q-1}{l}})+a_0$ with some positive integers r, l such that $l| (q-1)$. We call l the index of f(x) (see [1, 17]).

Let $Gal(\mathbb {F}_q/\mathbb {F}_p)$ be the Galois group of $\mathbb {F}_q$ over $\mathbb {F}_p$, it is clear that

$$Gal(\mathbb {F}_q/\mathbb {F}_p)=\langle \sigma \rangle =\{1,\sigma , \sigma ^2,\ldots , \sigma ^{m-1}\},$$

where $\sigma : c\mapsto c^p$ is an automorphism of $\mathbb {F}_q$, which is called Frobenius automorphism of $\mathbb {F}_q$ over $\mathbb {F}_p$. A well-known fact is that $\sigma (c)=c$ if and only if $c\in \mathbb {F}_p$. We extend $\sigma$ to a map $\tilde{\sigma }$ of $\mathbb {F}_q[x]$: $f(x) \mapsto (f(x))^p$, where f(x) is an arbitrary polynomial over $\mathbb {F}_q$.

In the following, we always assume that

$$\begin{aligned} f(x)=a_{r_k}x^{r_k}+a_{r_{k-1}}x^{r_{k-1}}+\ldots +a_{r_1}x^{r_1} +a_0\in \mathbb {F}_q[x],\end{aligned}$$

(2.1)

where $q-1\ge r_k>r_{k-1}>\cdots >r_1\ge 1$ and $a_{r_i}\ne 0$, $i=1,2,\ldots ,k$. Denote $r=r_1$. Then

$$f(x)=x^{r}h( x^{\frac{q-1}{l}})+a_0, $$

and the index l of f(x) is

$$ l=\frac{q-1}{\gcd ( r_k-r, r_{k-1}-r,\cdots , r_{2}-r, q-1)}. $$

Lemma 2.1

[18] Let $f(x)=x^{r}h( x^{\frac{q-1}{l}})\in \mathbb {F}_q[x]$ be a polynomial with the index l. Let $n_0=\sharp \{ i, 0\le i\le l-1: h(\zeta ^i)=0 \}$, where $\zeta$ is a primitive l-th root of unity in $\mathbb {F}_q$. For a nontrivial additive character $\psi$ of $\mathbb {F}_q$,

$$\begin{aligned}\bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x))-\frac{q}{l} n_0\bigg |\le (l-n_0) \cdot \gcd (r, \frac{q-1}{l})\cdot \sqrt{q}.\end{aligned}$$

Let f(x) be given in (2.1). Define a transformation of $\mathbb {F}_q[x]$:

$$\begin{aligned} \mathbb {F}_q[x]\longrightarrow & {} \mathbb {F}_q[x]\nonumber \\ \phi : f(x)\longmapsto & {} \phi (f(x)), \end{aligned}$$

(2.2)

where

$$\begin{aligned}\phi (f(x))= & {} \tilde{\sigma }^{v_{ k}}(a_{r_k}x^{r_k})+ \ldots + \tilde{\sigma }^{v_{ 1}}(a_{r_1}x^{r_1})+ \sigma ^{v_0}(a_0)\\ {}= & {} a_{r_k}^{p^{v_k}}x^{r_kp^{v_k}}+ \ldots + a_{r_1}^{p^{v_1}}x^{r_1p^{v_1}}+ a_0^{p^{v_0}},\end{aligned}$$

$0\le v_{ i}\le m-1, i=0, 1,2,\ldots ,k$.

For $x\in \mathbb {F}_q$, let ${\text {Tr}}_{q/p}(x)$ be the absolute trace of x over $\mathbb {F}_p$, then ${\text {Tr}}_{q/p}(\phi (f(x) ))={\text {Tr}}_{q/p}(f(x))$. For convenience, let $\psi$ be the canonical additive character of $\mathbb {F}_q$. Then

$$\begin{aligned} \sum _{x\in \mathbb {F}_q}\psi (f(x))= & {} \sum _{x\in \mathbb {F}_q}\zeta _p^{{\text {Tr}}_{q/p}( f(x)) }\nonumber \\ {}= & {} \sum _{x\in \mathbb {F}_q}\zeta _p^{{\text {Tr}}_{q/p}( \phi (f(x))) }\nonumber \\ {}= & {} \sum _{x\in \mathbb {F}_q}\psi ( \phi (f(x))). \end{aligned}$$

(2.3)

In fact, for an arbitrary nontrivial additive character of $\mathbb {F}_q$, (2.3) holds. Hence, Wu et al. [19] obtained the following result, which is an improvement of Lemma 2.1.

Lemma 2.2

[19] The notations are as above. Suppose that $l^*$ is the least index of $\phi (f(x))$ and $\phi (f(x))=x^{r^*}h(x^{\frac{q-1}{l^*}})$. Let $n_0=\sharp \{ i, 0\le i\le l^*-1: h({\zeta }^i)=0 \}$, where $\zeta$ is a primitive $l^*$ root of unity in $\mathbb {F}_q$. Then

$$\begin{aligned}\bigg |\sum _{x\in \mathbb {F}_q}\psi ( f(x) )-\frac{q}{l^*} n_0\bigg |\le (l^*-n_0)\cdot \gcd (r^*, \frac{q-1}{l^*})\cdot \sqrt{q}.\end{aligned}$$

In this paper, we shall generalize Lemma 2.2 to get improved bound for character sums of f(x). In order to investigate the bound for character sum of polynomial f(x), we consider the set of all positive divisors of $q-1$:

$$D=\{d\in \mathbb {N}: d|(q-1)\}. $$

Let $f(x)=\sum _{i=1}^ka_{r_i}x^{r_i}+a_0$ be defined as (2.1). Fix a $d\in D$ and $ds=q-1$. Then by division algorithm,

$$\begin{aligned} \begin{array}{ccc}r_i&{}=&{}sq_{0}^{(d,i)}+r_{0}^{(d,i)},\\ p r_i&{}=&{}sq_{1}^{(d,i)}+r_{1}^{(d,i)},\\ \cdots \\ p^{m-1}r_i&{}=&{}sq_{m-1}^{(d,i)}+r_{m-1}^{(d, i)}, \end{array}\end{aligned}$$

(2.4)

where $0\le r_{0}^{(d,i)},\ldots , r_{m-1}^{(d,i)}\le s-1$.

Let

$$\begin{aligned} R_{d, i}= & {} \{ r_{0}^{(d,i)}, r_{1}^{(d,i)},\ldots , r_{m-1}^{(d,i)}\}, i= 1,2,\ldots ,k,\\ r^{(d, i)}= & {} \min \{\gcd (s, r_j^{(d, i)}): 0\le j\le m-1\}, i=1,\ldots , k, \end{aligned}$$

and

$$\begin{aligned}T_d=\{r^{(d, i)}: i=1, \ldots , k\}.\end{aligned}$$

Without loss of generality, let $t=|T_d|$,

$$\begin{aligned} T_d=\{r^{(d, 1)},\ldots , r^{(d,t)}\}, \end{aligned}$$

(2.5)

and $r^{(d,1)}<r^{(d,2)}<\cdots < r^{(d,t)}$.

By (2.5), there exists a transformation $\phi _d$ of $\mathbb {F}_q[x]$ defined as (2.2) such that

$$\phi _d(f(x)) =\sum _{j=1}^t x^{r^{(d,j)}}h_j(x^s)+b,$$

where $b\in \mathbb {F}_q$ and $h_j(x^s)$ is a polynomial of $x^s$ over $\mathbb {F}_q$. In the following, we present our main result in Theorem 2.3.

Theorem 2.3

Let $d\in D$ and $sd=q-1$. Let $\gamma$ be a primitive element of the finite field $\mathbb {F}_q$ and $\zeta =\gamma ^s$ be a fixed primitive d-th root of unity in $\mathbb {F}_q$. Let f(x) be defined as (2.1) and $T_d$ be given by (2.5). Let $\phi _d$ be the transformation of $\mathbb {F}_q[x]$ such that $\phi _d( f(x))=\sum _{j=1}^{t } x^{r^{(d,j)}}h_j(x^s)+b$, where $b\in \mathbb {F}_q$ and $t=|T_d|$. Let

$$I_d= \{ i: 0\le i\le d-1 \text{ and } h_j(\zeta ^{i })\ne 0 \text{ for } \text{ some } j, 1\le j\le t\} $$

and $n_{d,0}=d-|I_d|$. Then for each $d\in D$,

$$\bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x))\bigg |\le (d -n_{d,0})\sqrt{q}(r^{(d,t)}-\frac{1}{d})+\frac{q}{d} n_{d,0}.$$

Moreover, if d runs over all positive factors of $q-1$, then

$$\bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x))\bigg |\le \min _{d\in D}\{ (d -n_{d,0})\sqrt{q}(r^{(d,t)}-\frac{1}{d})+\frac{q}{d} n_{d,0}\}.$$

Proof

Take a $d\in D$ and $ds=q-1$, let

$$\phi _d(f(x))=\sum _{j=1}^{t } x^{r^{(d,j)}}h_j(x^s)+b,$$

$C_0=\langle \gamma ^d \rangle$ be a subgroup of $\mathbb {F}_q^*$ of index d, and $\mathbb {F}_q^*=\cup _{i=0}^{d-1}\gamma ^i C_0$.

Let

$$I_d= \{ i: 0\le i\le d-1 \text{ and } h_j(\zeta ^{i })\ne 0 \text{ for } \text{ some } j, 1\le j\le t \} $$

be a subset of $\{0,1,\ldots , d-1\}$. Then

$$\begin{aligned}{} & {} \sum _{x\in \mathbb {F}_q} \psi (\phi _d(f(x))-b) \\= & {} 1+\sum _{i=0}^{d-1}\sum _{x\in C_0}\psi (\phi _d(f(\gamma ^ix))-b) \\ {}= & {} 1+\frac{1}{d}\sum _{i=0}^{d-1}\sum _{x\in \mathbb {F}_q^*}\psi (\phi _d(f(\gamma ^ix^d))-b) \\ {}= & {} 1+\frac{1}{d}\sum _{i=0}^{d-1}( \sum _{x\in \mathbb {F}_q^*}\psi (\sum _{j=1}^{t } x^{dr^{(d,j)}}\gamma ^{ir^{(d,j)}}h_j(\zeta ^i))) \\ {}= & {} 1+\frac{1}{d}\sum _{i\in I_d}( \sum _{x\in \mathbb {F}_q^*}\psi (\sum _{j=1}^{t } x^{dr^{(d,j)}}\gamma ^{ir^{(d,j)}}h_j(\zeta ^i))) + \frac{q-1}{d} n_{d,0} \\ {}= & {} \frac{1}{d}\sum _{i\in I_d}( \sum _{x\in \mathbb {F}_q}\psi (\sum _{j=1}^{t } x^{dr^{(d,j)}}\gamma ^{ir^{(d,j)}}h_j(\zeta ^i))) +\frac{q}{d} n_{d,0}.\end{aligned}$$

Furthermore,

$$\begin{aligned}{} & {} \bigg |\sum _{x\in \mathbb {F}_q}\psi (\phi _d(f(x))-b)\bigg |\nonumber \\ {}\le & {} \frac{1}{d}\sum _{i\in I_d}\bigg | (\sum _{x\in \mathbb {F}_q}\psi (\sum _{j=1}^{t}x^{dr^{(d,j)}}\gamma ^{ir^{(d,j)}}h_j(\zeta ^i)))\bigg | +\frac{q}{d} n_{d,0}\nonumber \\ {}\le & {} \frac{d-n_{d,0}}{d} \{\gcd (dr^{(d,t)}, q-1)-1\}\sqrt{q}+\frac{q}{d} n_{d,0}\nonumber \\\le & {} (d -n_{d,0})\sqrt{q} (r^{(d,t)}-\frac{1}{d})+\frac{q}{d} n_{d,0}, \end{aligned}$$

(2.6)

note that

$$\psi (\phi _d(f(x))-b)= \psi (-b)\psi (\phi _d(f(x))),$$

where $|\psi (-b)|=1$. Recall that $\sum _{x\in \mathbb {F}_q}\psi (f(x))=\sum _{x\in \mathbb {F}_q}\psi ( \phi (f(x)))$, then

$$\bigg |\sum _{x\in \mathbb {F}_q} \psi (f(x)) \bigg |\le (d -n_{d,0})\sqrt{q}(r^{(d,t)}-\frac{1}{d})+\frac{q}{d} n_{d,0}.$$

Moreover, if d runs over all positive divisors of $q-1$, then

$$\begin{aligned} \bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x)) \bigg |\le \min _{d\in D }\{ (d -n_{d,0})\sqrt{q}(r^{(d,t)}-\frac{1}{d})+\frac{q}{d} n_{d,0}\}.\end{aligned}$$

$\square$

In fact, we may obtain an improved upper bound than one in (2.6). In Theorem 2.3, if $d\in D$ is fixed and $|T_d|=1$, then a generalized form of Lemma 2.2 in [19] is obtained immediately.

Corollary 2.4

Let f(x) be defined as (2.1), d, s two positive integers with $ds=q-1$, and $T_d=\{r^{(d,1)}\}$ with $r^{(d,1)}>0$. Let $\phi _d$ be the transformation of $\mathbb {F}_q[x]$ such that $\phi _d( f(x))= x^{r^{(d,1)}}h(x^s)$. Let $\gamma$ be a primitive element of the finite field $\mathbb {F}_q$ and $\zeta =\gamma ^s$ be a fixed primitive d-th root of unity in $\mathbb {F}_q$. Let $I_{d}=\{ i: 0\le i\le d-1 \text{ and } h(\zeta ^i)=0 \}$ and $n_{d,0}=|I_d|$. Then

$$\begin{aligned} \bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x))-\frac{q}{d} n_{d,0}\bigg |\le (d-n_{d,0})(r^{(d,1)}-\frac{1}{d})\sqrt{q}. \end{aligned}$$

Remark 2.5

In Theorem 2.3, the Weil bound of character sum associated to an arbitrary polynomial f(x) depends on $d\in D, n_{d,0}$, and $r^{(d,t)}$. The Weil bound in Theorem 2.3 is much better than the ones of Lemmas 2.1 and 2.2. In fact, the bound for character sum of f(x) in Lemma 2.2 presented by Wu et al. in [19] is a special case of Corollary 2.4.

In the following, we give some examples to show that the Weil bound for character sums of f(x) in Theorem 2.3 is indeed an improved bound compared to the bounds in (1.1), Lemmas 2.1 and 2.2.

Example 2.6

Let $q=5^2$ and $f(x)=x^{19}+x^{11}+x^5\in \mathbb {F}_q[x]$. The Weil bound in (1.1) is trivial because of the high degree. The bound in Lemma 2.1 is also trivial since the index of f(x) is 12.

In Lemma 2.2, the least index of $\phi (f(x))$ is

$$d^*=\frac{24}{\gcd (23-5,11-5,24)}=4$$

and $\phi (f(x))=x^{5}((x^{6})^3+x^6+1)$. Then the bound for character sums of f(x) in Lemma 2.2 is

$$ |\sum _{x\in \mathbb {F}_q}\psi (f(x)) |\le 20.$$

In Theorem 2.3, $D=\{d: d|24\}=\{ 2,3,4,6,8,12\}$.

(i) For $d=2$, we have $R_{2,1}=\{1,5\}$ and $R_{2,2}=R_{2,3}=\{7,11\}$. Let $T_2=\{1,7\}$, there exists the transformation $\phi _2$ of $\mathbb {F}_q[x]$ such that $\phi _2(f(x))=x^7((x^{12})^4+x^{12})+x(x^{12})^2$, then $|\sum _{x\in \mathbb {F}_q}\psi (f(x)) |\le 10$.

(ii) For $d=3$, we have $R_{3,1}= R_{3,2}=R_{3,3}=\{1,5\}$. Let $T_3=\{5\}$, there exists the transformation $\phi _3$ of $\mathbb {F}_q[x]$ such that $\phi _3(f(x))=x^5((x^6)^{15}+(x^6)+1)$, then $|\sum _{x\in \mathbb {F}_q}\psi (f(x)) |\le 20$.

(iii) For $d=4$, we have $R_{4,1}=\{1,5\}$ and $R_{4,2}=R_{4,3}=\{3,7\}$. Let $T_4=\{1,3\}$, there exists the transformation $\phi _4$ of $\mathbb {F}_q[x]$ such that $\phi _4(f(x))=x^3((x^8)^{2}+(x^8)) +x((x^8)^3)$, then $|\sum _{x\in \mathbb {F}_q}\psi (f(x)) |\le 15$. Similarly, for other $d\in D$, the bounds of f(x) are larger than the above cases. Hence by Theorem 2.3, the bound for character sums of f(x) is

$$\bigg |\sum _{x\in \mathbb {F}_q}\psi (f(x)) \bigg |\le \min \{ 10,15,20\}=10.$$

Example 2.7

Let $f(x)=x^{19}+ax^4\in \mathbb {F}_{27}[x]$, where $a\in \mathbb {F}_{27}^*$. The bounds given by (1.1), Lemmas 2.1 and 2.2 are all trivial. Another upper bound for exponential sums of f(x) presented by Wu et al. in [19, Example 1.5] is $|\sum _{x\in \mathbb {F}_{27}} \psi (f(x)) |\le 4\sqrt{27}$.

In Theorem 2.3, take $d=2$, we have $R_{2,1}=\{ 4,12,10\}$, $R_{2,2}=\{ 6,5,2\}$, and $T_2=\{4,2\}$, there exists the map $\phi _2$ of $\mathbb {F}_{27}[x]$ such that $\phi _2(f(x))=x^2((x^{s})^{13} )+cx^4$, by Theorem 2.3, $|\sum _{x\in \mathbb {F}_{27}}\psi (f(x)) |\le 2\sqrt{27}$.

Example 2.8

Let $f(x)=x^{10}+ax^5\in \mathbb {F}_{27}[x]$, where $a\in \mathbb {F}_{27}^*$. The bounds given by (1.1), Lemmas 2.1 and 2.2 are all trivial. Another upper bound for exponential sums of f(x) presented by Wu et al. in [19, Example 1.6] is $|\sum _{x\in \mathbb {F}_{27}} \psi (f(x)) |\le 4\sqrt{27}$.

In Theorem 2.3, take $d=2$, we have $R_{2,1}=\{ 5,2,4\}$, $R_{2,2}=\{ 10,4,6\}$, and $T_2=\{4\}$, there exists the map $\phi _2$ of $\mathbb {F}_q[x]$ such that $\phi _2(f(x))=x^4( (x^{s})^{2} +c(x^{s})^{2})$, by Theorem 2.3, $|\sum _{x\in \mathbb {F}_q}\psi (f(x)) |\le 2\sqrt{27}$.

From the above examples one sees that the bound we obtained is indeed an improvement compared to that Wan and Wu et al. in [18] and [19].

3 Concluding remarks

This paper mainly studied the Weil bounds for character sums associated to an arbitrary polynomial f(x) over a finite field $\mathbb {F}_q$. The result of Theorem 2.3 presented an improved upper bound than the results of Wan and Wu et al. in [18] and [19], respectively.