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

TL;DR

This paper establishes criteria for identifying polynomials capable of generating Sidon sets over finite fields, proving certain classes cannot, and linking these polynomials to planar polynomials, advancing understanding of Sidon set construction.

Contribution

It introduces new criteria for determining Sidon-generating polynomials and proves non-existence results for specific polynomial classes, connecting to planar polynomials.

Findings

Certain monomials and cubic polynomials cannot generate Sidon sets over _p

Derived criteria for identifying Sidon-generating polynomials

Connected Sidon-generating polynomials to planar polynomials

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.