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

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-16@?+1)(p+2@?p-16@?)/2~2p^2/9 pairs (a,b)@?F"p^2 with the property that f(x)+ag(x)+bx is a permutation polynomial, then there exist elements c,d,e@?F"p with the property that f(x)=cg(x)+dx+e.