$Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$$

Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$

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 pol...

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Veröffentlicht in:	arXiv.org 2023-06
Hauptverfasser:	Afifurrahman, Muhammad, Barra, Aleams
Format:	Artikel
Sprache:	eng
Schlagworte:	Alliances Criteria Polynomials
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creator	Afifurrahman, Muhammad Barra, Aleams
description	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.
doi_str_mv	10.48550/arxiv.2111.08886
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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.</description><identifier>EISSN: 2331-8422</identifier><identifier>DOI: 10.48550/arxiv.2111.08886</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Alliances ; Criteria ; Polynomials</subject><ispartof>arXiv.org, 2023-06</ispartof><rights>2023. 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title	Determining Sidon Polynomials on Sidon Sets over $\mathbb{F}_q\times \mathbb{F}_q$
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Determining Sidon Polynomials on Sidon Sets over \(\mathbb{F}_q\times \mathbb{F}_q\)