On the graph of a function over a prime field whose small powers have bounded degree

Let f be a function from a finite field F"p with a prime number p of elements, to F"p. In this article we consider those functions f(X) for which there is a positive integer n>2p-1-114 with the property that f(X)^i, when considered as an element of F"p[X]/(X^p-X), has degree at most p-2-n+i, for all i=1,...,n. We prove that every line is incident with at most t-1 points of the graph of f, or at least n+4-t points, where t is a positive integer satisfying n>(p-1)/t+t-3 if n is even and n>(p-3)/t+t-2 if n is odd. With the additional hypothesis that there are t-1 lines that are incident with at least t points of the graph of f, we prove that the graph of f is contained in these t-1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t-1 and prove the conjecture for t=2 and t=3. These results apply to functions that determine less than p-2p-1+114 directions. In particular, the proof of the conjecture for t=2 and t=3 gives new proofs of the result of Lovasz and Schrijver [L. Lovasz, A. Schrijver, Remarks on a theorem of Redei, Studia Sci. Math. Hungar. 16 (1981) 449-454] and the result in [A. Gacs, On a generalization of Redei's theorem, Combinatorica 23 (2003) 585-598] respectively, which classify all functions which determine at most 2(p-1)/3 directions.