A degree bound for planar functions

TL;DR

This paper establishes a degree bound for planar functions over finite fields using Gauss sums, leading to new classifications of planar monomials for fields of characteristic greater than 5.

Contribution

It introduces a novel degree bound for compositions with planar functions, completing classifications for certain finite fields and proposing a conjecture for full classification.

Findings

Proves a degree bound for planar functions using Stickelberger's theorem.

Completes classification of planar monomials for fields with degree powers of two and characteristic >5.

States a conjecture linking digit sums to the classification of planar monomials.

Abstract

Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} d_{\mathsf{alg}}(G \circ F) - d_{\mathsf{alg}}(G) \le \frac{n(p-1)}{2}, \end{equation*} where $q = p^n$ with $p$ a prime and $n$ a positive integer, and $d_{\mathsf{alg}}(F)$ is the algebraic degree of $F$, i.e., the maximum degree of the corresponding system of $n$ lowest-degree interpolating polynomials for $F$ considered as a function on $\mathbb{F}_p^n$. This bound implies the (known) classification of planar polynomials over $\mathbb{F}_p$ and planar monomials over $\mathbb{F}_{p^2}$. As a new result, using the same degree bound, we complete the classification of planar monomials for all $n = \smash{2^k}$ with $p>5$ and $k$ a non-negative integer. Finally, we state a conjecture on the sum of the base-$p$ digits of integers modulo $q-1$ that implies the complete classification of planar monomials over finite fields of characteristic $p>5$.