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

AbstractA 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−16⌉+1)(p+2⌈p−16⌉)/2≈2p2/9 pairs (a,b)∈Fp2 with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e∈Fp with the property that f(x)=cg(x)+dx+e