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

Abstract

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.

Submitted