Low-degree planar polynomials over finite fields of characteristic two

Planar functions are mappings from a finite field Fq to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q1/4 that induce planar functions on Fq, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over Fq that induce planar functions on infinitely many extensions of Fq. The proof strategy is to study the number of Fq-rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions.