Combinatorial Theory

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 \(p›5\).

Mathematics Subject Classifications: 05B25, 11T06, 11T24

Keywords: Planar function, algebraic degree, Stickelberger's theorem, digit sum