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\rightarrow\F_q is called homogenous if $f(\lambda x)=\lambda^df(x)$ for all \lambda\in\F_p and x\in\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.Comment: Introduction modified to: 1. give the correct definition of equivalence, 2. add some references. Other part unaltere