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.