How to use Rédei polynomials in higher dimensional spaces

A polynomial with coefficients from a finite field GF (q) that is the product of linear polynomials is usually referred to as a Rédei polynomial. This is due to the appearance of the polynomial R(T, S) = (x,y)∈A (T − xS + y), where A is some subset of GF (q)2, in the book of Rédei [15] from the early seventies. In the affine plane AG(2, q) the point (x, y) is incident with the line Y = mX + α if and only if α = −mx + y. Hence −α is a root of R(T,m) of multiplicity k if and only if the line Y = mX + α is incident with k points of the set A. This observation allows one to look at a problem of the following type: Given a subset of points of the affine plane that has restricted intersection properties with the lines of the plane, say something about the size of the subset or, clearly better, determine the possibilities for such a subset