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 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.