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Volume 38, Issue 2
A Recursive Formula and an Estimation for a Specific Exponential Sum

Xiwang Cao & Liqin Qian

DOI: 10.4208/cmr.2021-0030

Commun. Math. Res., 38 (2022), pp. 184-205.

Published online: 2022-02

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  • Abstract

Let $\mathbb{F}_q$ be a finite field and $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$. Let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with ${\rm gcd}(n,q) = 1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}} \chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with a $a≠0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^∗_{q^s}}\chi^{(s)} (ac^u+bc^{−1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided too.

  • Keywords

Exponential sums, finite fields, Dickson polynomials, sequences.

  • AMS Subject Headings

11T23

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-38-184, author = {Cao , Xiwang and Qian , Liqin}, title = {A Recursive Formula and an Estimation for a Specific Exponential Sum}, journal = {Communications in Mathematical Research }, year = {2022}, volume = {38}, number = {2}, pages = {184--205}, abstract = {

Let $\mathbb{F}_q$ be a finite field and $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$. Let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with ${\rm gcd}(n,q) = 1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}} \chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with a $a≠0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^∗_{q^s}}\chi^{(s)} (ac^u+bc^{−1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided too.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2021-0030}, url = {http://global-sci.org/intro/article_detail/cmr/20270.html} }
TY - JOUR T1 - A Recursive Formula and an Estimation for a Specific Exponential Sum AU - Cao , Xiwang AU - Qian , Liqin JO - Communications in Mathematical Research VL - 2 SP - 184 EP - 205 PY - 2022 DA - 2022/02 SN - 38 DO - http://doi.org/10.4208/cmr.2021-0030 UR - https://global-sci.org/intro/article_detail/cmr/20270.html KW - Exponential sums, finite fields, Dickson polynomials, sequences. AB -

Let $\mathbb{F}_q$ be a finite field and $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$. Let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with ${\rm gcd}(n,q) = 1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}} \chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with a $a≠0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^∗_{q^s}}\chi^{(s)} (ac^u+bc^{−1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided too.

Cao , Xiwang and Qian , Liqin. (2022). A Recursive Formula and an Estimation for a Specific Exponential Sum. Communications in Mathematical Research . 38 (2). 184-205. doi:10.4208/cmr.2021-0030
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