Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913558345809920 |
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| author | Afifurrahman, Muhammad Barra, Aleams |
| author_facet | Afifurrahman, Muhammad Barra, Aleams |
| contents | Let $p$ be a prime, and $q=p^n$ be a prime power. In his works on Sidon sets over $\mathbb{F}_q\times \mathbb{F}_q$, Cilleruelo conjectured about polynomials that could generate $q$-element Sidon sets over $\mathbb{F}_q\times \mathbb{F}_q$.
Here, we derive some criteria for determining polynomials that could generate $q$-element Sidon set over $\mathbb{F}_q\times \mathbb{F}_q$. Using these criteria, we prove that certain classes of monomials and cubic polynomials over $\mathbb{F}_p$ cannot be used to generate $p$-element Sidon set over $\mathbb{F}_p\times \mathbb{F}_p$. We also discover a connection between the needed polynomials and planar polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_08886 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$ Afifurrahman, Muhammad Barra, Aleams Combinatorics Number Theory 11T06 (primary), 11B83, 05A20 (secondary) Let $p$ be a prime, and $q=p^n$ be a prime power. In his works on Sidon sets over $\mathbb{F}_q\times \mathbb{F}_q$, Cilleruelo conjectured about polynomials that could generate $q$-element Sidon sets over $\mathbb{F}_q\times \mathbb{F}_q$. Here, we derive some criteria for determining polynomials that could generate $q$-element Sidon set over $\mathbb{F}_q\times \mathbb{F}_q$. Using these criteria, we prove that certain classes of monomials and cubic polynomials over $\mathbb{F}_p$ cannot be used to generate $p$-element Sidon set over $\mathbb{F}_p\times \mathbb{F}_p$. We also discover a connection between the needed polynomials and planar polynomials. |
| title | Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$ |
| topic | Combinatorics Number Theory 11T06 (primary), 11B83, 05A20 (secondary) |
| url | https://arxiv.org/abs/2111.08886 |