On homogeneous planar functions

Abstract

Let p be an odd prime and F q be the finite field with q = p n elements. A planar function f : F q → F q is called homogenous if f (λx) = λ d f (x) for all λ ∈ F p and x ∈ F q , where d is some fixed positive integer. We characterize x 2 as the unique homogenous planar function over F p 2 up to equivalence.