On homogeneous planar functions

We introduce the notion of homogeneous planar functions, and characterize x2 as the unique homogeneous planar function over finite fields with prime square elements. To be specific, we show that if f is a planar function defined over Fp2 such that f(λx)=λdf(x) for all λ∈Fp and x∈Fp2, with p an odd prime and d a fixed integer, then f is equivalent to x2.