Crooked maps in

A map is called crooked if the set is an affine hyperplane for every fixed (where is considered as a vector space over ). We prove that the only crooked power maps are the quadratic maps x2i+2j with gcd(n,i−j)=1. This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0 d pn−2 the set contains n linearly independent elements, where γ and a≠0 are arbitrary elements from .