Some planar monomials in characteristic 2

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They were originally defined only in odd characteristic, but recently Zhou introduced a definition in even characteristic which yields similar applications. In this paper we show that certain functions over $\mathbb{F}_{2^r}$ are planar, which proves a conjecture of Schmidt and Zhou. The key to our proof is a new result about the $\mathbb{F}_{q^3}$-rational points on the degree-$(q-1)$ Fermat curve $x^{q-1}+y^{q-1}=z^{q-1}$.