A degree bound for planar functions

TL;DR

This work proves a degree-growth bound for planar functions: for a planar $F$ on $\mathbb{F}_q$ and any nonzero $G$, the algebraic degree grows additively under composition, satisfying $d_{\mathsf{alg}}(G\circ F) - d_{\mathsf{alg}}(G) \le \frac{n(p-1)}{2}$ with $q=p^n$. The proof combines additive and multiplicative $p$-adic Fourier analysis, Stickelberger’s theorem on Gauss sums, and a careful change-of-basis analysis to relate Walsh-transform coordinates to interpolating-polynomial degrees. As consequences, the result recovers the known classifications of planar polynomials over $\mathbb{F}_p$ and planar monomials over $\mathbb{F}_{p^2}$, and, for $p>5$, yields a complete classification of planar monomials over $\mathbb{F}_{p^{2^k}}$; a conjecture on base-$p$ digit sums is proposed, which would imply the full monomial classification for all $p>5$. The approach highlights a structural link between additive geometry and multiplicative digit-structure, offering a new pathway to understanding planarity and its obstructions in finite fields.

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 $p>5$.

Theorems & Definitions (39)

• theorem 1
• Corollary 1
• theorem 2
• remark 1
• remark 2
• definition 1: Fourier transformation
• theorem 3: Beyne2023
• proof
• theorem 4: Stickelberger's theorem for Gauss sums lang2012cyclotomic
• lemma 1