On the Classification of Exceptional Planar Functions over $\mathbb{F}_{p}$

TL;DR

This paper advances the classification of exceptional planar functions over finite fields using algebraic geometry tools, providing partial results and insights into their structure.

Contribution

It introduces new partial classification results for exceptional planar monomials over finite fields employing Weil, Bezout, and Bertini theorems.

Findings

Partial classification of exceptional planar monomials achieved.

Application of algebraic geometry techniques to finite field functions.

Identification of conditions under which monomials are exceptional planar.

Abstract

We will present many strong partial results towards a classification of exceptional planar/PN monomial functions on finite fields. The techniques we use are the Weil bound, Bezout's theorem, and Bertini's theorem.