On the number of directions determined by a pair of functions over a prime field

Abstract

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 ⌈ p − 1 6 ⌉ + 1 ) ( p + 2 ⌈ p − 1 6 ⌉ ) / 2 ≈ 2 p 2 / 9 pairs ( a , b ) ∈ F p 2 with the property that f ( x ) + a g ( x ) + b x is a permutation polynomial, then there exist elements c , d , e ∈ F p with the property that f ( x ) = c g ( x ) + d x + e .