Primitive pairs of rational functions with prescribed traces over finite fields
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916493185253376 |
|---|---|
| author | Nath, Shikhamoni Basnet, Dhiren Kumar |
| author_facet | Nath, Shikhamoni Basnet, Dhiren Kumar |
| contents | Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $ε\in \mathbb{F}_{q^m}$, such that $(f(ε), g(ε))$ is a primitive pair in $\mathbb{F}_{q^m}$ with two prescribed traces, $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(ε)=a$ and $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(ε^{-1})=b$, where $f(x), g(x) \in \mathbb{F}_{q^m}(x)$ are rational functions with some restrictions and $a, b \in \mathbb{F}_q$. Also, we show that there exists an element $ε\in \mathbb{F}_{q^m}$ satisfying our desired properties in all but finitely many fields $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$. We also calculate possible exceptional pairs explicitly for $m\geq 9$, when degree sums of both the rational functions are taken to be 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive pairs of rational functions with prescribed traces over finite fields Nath, Shikhamoni Basnet, Dhiren Kumar Number Theory 12E20, 11T23 Let $q$ be a positive integral power of some prime $p$ and $\mathbb{F}_{q^m}$ be a finite field with $q^m$ elements for some $m \in \mathbb{N}$. Here we establish a sufficient condition for the existence of a non-zero element $ε\in \mathbb{F}_{q^m}$, such that $(f(ε), g(ε))$ is a primitive pair in $\mathbb{F}_{q^m}$ with two prescribed traces, $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(ε)=a$ and $\Tr_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(ε^{-1})=b$, where $f(x), g(x) \in \mathbb{F}_{q^m}(x)$ are rational functions with some restrictions and $a, b \in \mathbb{F}_q$. Also, we show that there exists an element $ε\in \mathbb{F}_{q^m}$ satisfying our desired properties in all but finitely many fields $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$. We also calculate possible exceptional pairs explicitly for $m\geq 9$, when degree sums of both the rational functions are taken to be 3. |
| title | Primitive pairs of rational functions with prescribed traces over finite fields |
| topic | Number Theory 12E20, 11T23 |
| url | https://arxiv.org/abs/2411.15568 |