A three-dimensional analogue of the classical direction problem is proposed and an asymptotically sharp bound for the number of directions determined by a non-planar set in AG (3, p), p prime, is proved. Using the terminology of permutation polynomials the main result states that if there are more than (2 ⌈ frac(p - 1, 6) ⌉ + 1) (p + 2 ⌈ frac(p - 1, 6) ⌉) / 2 ≈ 2 p2 / 9 pairs (a, b) ∈ Fp2 with the property that f (x) + a g (x) + b x is a permutation polynomial, then there exist elements c, d, e ∈ Fp with the property that f (x) = c g (x) + d x + e. © 2007 Elsevier Inc. All rights reserved.
Ball, S., Gács, A., & Sziklai, P. (2008). On the number of directions determined by a pair of functions over a prime field. Journal of Combinatorial Theory. Series A, 115(3), 505–516. https://doi.org/10.1016/j.jcta.2007.08.001