$F$-Diophantine sets over finite fields
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909603723214848 |
|---|---|
| author | Yip, Chi Hoi Yoo, Semin |
| author_facet | Yip, Chi Hoi Yoo, Semin |
| contents | Let $k \geq 2$, $q$ be an odd prime power, and $F \in \mathbb{F}_q[x_1, \ldots, x_k]$ be a polynomial. An $F$-Diophantine set over a finite field $\mathbb{F}_q$ is a set $A \subset \mathbb{F}_q^*$ such that $F(a_1, a_2, \ldots, a_k)$ is a square in $\mathbb{F}_q$ whenever $a_1, a_2, \ldots, a_k$ are distinct elements in $A$. In this paper, we provide a strategy to construct a large $F$-Diophantine set, provided that $F$ has a nice property in terms of its monomial expansion. In particular, when $F=x_1x_2\ldots x_k+1$, our construction gives a $k$-Diophantine tuple over $\mathbb{F}_q$ with size $\gg_k \log q$, significantly improving the $Θ((\log q)^{1/(k-1)})$ lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00310 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $F$-Diophantine sets over finite fields Yip, Chi Hoi Yoo, Semin Number Theory 11D79, 11T06, 11T24 Let $k \geq 2$, $q$ be an odd prime power, and $F \in \mathbb{F}_q[x_1, \ldots, x_k]$ be a polynomial. An $F$-Diophantine set over a finite field $\mathbb{F}_q$ is a set $A \subset \mathbb{F}_q^*$ such that $F(a_1, a_2, \ldots, a_k)$ is a square in $\mathbb{F}_q$ whenever $a_1, a_2, \ldots, a_k$ are distinct elements in $A$. In this paper, we provide a strategy to construct a large $F$-Diophantine set, provided that $F$ has a nice property in terms of its monomial expansion. In particular, when $F=x_1x_2\ldots x_k+1$, our construction gives a $k$-Diophantine tuple over $\mathbb{F}_q$ with size $\gg_k \log q$, significantly improving the $Θ((\log q)^{1/(k-1)})$ lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma. |
| title | $F$-Diophantine sets over finite fields |
| topic | Number Theory 11D79, 11T06, 11T24 |
| url | https://arxiv.org/abs/2406.00310 |