On the classification of planar monomials over fields of square order

Abstract Let F q be a finite field of characteristic p and F q [ X ] denote the ring of polynomials in X over F q . A polynomial f ∈ F q [ X ] is called a permutation polynomial over F q if f induces a bijection of F q under substitution. A polynomial f ∈ F q [ X ] is said to be planar over F q if for every non-zero a ∈ F q , the polynomial f ( X + a ) − f ( X ) is a permutation polynomial over F q . 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 p 2 . In this article we study planar monomials over fields of square order, obtaining a complete classification of planar monomials over fields of order p 4 .