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 Fq, then for all non-zero functions G : Fq → Fq, we have {equation*} deg \ G ∘ F - deg \ G ≤ n(p-1)/2\,, {equation*} where q = pⁿ with p a prime and n a positive integer, and deg \ F is the algebraic degree of F, i.e., the degree of the corresponding multivariate polynomial over Fₚ. This bound leads to a simpler proof of the classification of planar polynomials over Fₚ and planar monomials over Fp². As a new result, using the same degree bound, we complete the classification of planar monomials for all n = 2ᵏ 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 $p>5$.