A degree bound for planar functions

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_{$\mathsf{a}\mathsf{l}\mathsf{g}$}(G \circ F) - d_{$\mathsf{a}\mathsf{l}\mathsf{g}$}(G) \le \frac{n(p-1)}{2}, \end{equation*} where $q=p^n$ with $p$ a prime and $n$ a positive integer, and $d_{\mathsf{a}\mathsf{l}\mathsf{g}}(F)$ 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 $n=2^k$ 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 $p>5$.