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 F₏ℂ. 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.