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 \(F_q\), then for all non-zero functions \(G : F_q → F_q\), we have {equation*} dalg(G ∘ F) - dalg(G) ≤ n(p-1)/2, {equation*} where \(q = p^n\) with \(p\) a prime and \(n\) a positive integer, and \(dalg(F)\) 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 \(F_p^n\). This bound implies the (known) classification of planar polynomials over \(F_p\) and planar monomials over \(Fp^2\). 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 \(p›5\).