A degree bound for planar functions

2024/07/05 by Beierle, Christof, Beyne, Tim
#05B25 #11T06 #11T24 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.04570

Abstract

Using Stickelberger's theorem on Gauss sums, we show that if F is a planar function on a finite field \mathbbF_q, then for all non-zero functions G : \mathbbF_q → \mathbbF_q, we have d_alg(G ∘ F) - d_alg(G) ≤ (n(p-1))/(2), where q = pⁿ with p a prime and n a positive integer, and d_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 \mathbbF_pⁿ. This bound implies the (known) classification of planar polynomials over \mathbbF_p and planar monomials over \mathbbF_p². As a new result, using the same degree bound, we complete the classification of planar monomials for all n = \smash2^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.

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