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 $p>5$.
Categories: math.CO