On the classification of planar monomials over fields of square order

Let Fq be a finite field of characteristic p and Fq[X] denote the ring of polynomials in X over Fq. A polynomial f∈Fq[X] is called a permutation polynomial over Fq if f induces a bijection of Fq under substitution. A polynomial f∈Fq[X] is said to be planar over Fq if for every non-zero a∈Fq, the polynomial f(X+a)−f(X) is a permutation polynomial over Fq. Planar polynomials have only been classified over prime fields, whereas the problem of classifying planar monomials has only been completely resolved over fields of order p and p2. In this article we study planar monomials over fields of square order, obtaining a complete classification of planar monomials over fields of order p4.