Some planar monomials in characteristic 2

TL;DR

This paper proves a conjecture by demonstrating that specific functions over finite fields of characteristic 2 are planar, thereby expanding the class of known planar functions and their applications in combinatorics.

Contribution

It introduces new planar functions in characteristic 2 and establishes their properties, confirming a conjecture of Schmidt and Zhou.

Findings

Certain functions over _{2^r} are planar.

A new result on _{q^3}-rational points on Fermat curves.

Extension of planar function theory to even characteristic.

Abstract

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 $x^{q-1}+y^{q-1}=z^{q-1}$.