Determining Sidon polynomials on Sidon sets over finite fields

Let p be a prime, and q=pⁿ be a prime power. In his works on Sidon sets over Fq × Fq, Cilleruelo conjectured about polynomials that could generate q-element Sidon sets over Fq× Fq. Here, we derive some criteria for determining polynomials that could generate q-element Sidon set over Fq× Fq. Using these criteria, we prove that certain classes of monomials and cubic polynomials over Fₚ cannot be used to generate p-element Sidon set over Fₚ× Fₚ. We also discover a connection between the needed polynomials and planar polynomials.