Low-degree planar monomials in characteristic two

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on F_q of the form c-->uc^t, where q is a power of 2, t is an integer with 0<t<=q^{1/4}, and u is a nonzero element of F_q. This settles and sharpens a conjecture of Schmidt and Zhou.