The classification of planar monomials over fields of prime square order

Abstract

Abstract. Planar functions were introduced by Dembowski and Ostrom in 1968 to describe affine planes possessing collineation groups with particular properties. To date their classification has only been resolved for functions over fields of prime order. In this article we classify planar monomials over fields of order p 2 with p a prime. Preliminary results and notationLet p be a prime, e a natural number, q = p e and let F q denote the finite field of order q. The ring of polynomials in X over F q is denoted by F q [X]. It is well known that any function over F q can be uniquely represented by a polynomial inis called a permutation polynomial over F q if f induces a bijection of F q under substitution. We will use the classical criterion of Hermite for a polynomial to be a permutation polynomial. is a permutation polynomial over F q if and only if(i) f has exactly one root in F q , and (ii) the reduction of f t mod (X q − X), with 0 < t < q − 1 and t ≡ 0 mod p, has degree less than q − 1.A polynomial f ∈ F q [X] is called a planar polynomial if for every non-zero a ∈ F q , the polynomial f (X + a) − f (X) is a permutation polynomial over F q . Any quadratic polynomial is planar over any field of odd characteristic. It is also straightforward to show that there are no planar polynomials over fields of characteristic 2. Consequently all statements given in this article concern fields of odd prime characteristic p.The motivation for studying planar functions stems from an article of Dembowski and Ostrom [3], where they used such functions to construct affine planes of order n with collineation groups of order n 2 . In addition, it was recently shown that the classification of planar polynomials of the form e−1 i,j=0

Planar functions and perfect nonlinear monomials over finite fields

“…A folk conjecture in the subject asserts that there are no further examples. This is known to be true for r = 1 [13] and r = 2 [4], and also for r = 4 if p > 3 [5]. However, as noted in [3], the methods used in these papers will likely not extend to much larger values of r. In this paper we prove this conjecture for all large r: Date: January 21, 2013.…”

Classification of Planar Monomials Over Finite Fields of Small Order

“…For an odd prime p, the classification of planar monomials over the fields F p , F p 2 , F p 3 , and F p 4 is known from the works [13], [6], [2], and [7], respectively. Theorem 3 ([13, 6, 2, 7]).…”

On homogeneous planar functions

“…It is natural to consider the classification of all planar functions over F p 2 , but this turns out to be very difficult. The proof of [8] makes use of Hermite criterion for permutation polynomials, which has been followed by [3,4] to classify all planar monomials over F p 2 and F p 4 . Recently, all planar functions over F q of the form x n with (n − 1) 4 ≤ q have been classified in [24].…”