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 Fp with a prime number p of elements, to Fp. In this article we consider those functions f(X) for which there is a positive integer n > 2 √ p− 1− 11 4 with the property that f(X) , when considered as an element of Fp[X]/(X −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−2 √ p− 1+ 11 4 directions. In particular, the proof of the conjecture for t = 2 and t = 3 gives new proofs of the result of Lovász and Schrijver [7] and the result in [5] respectively, which classify all functions which determine at most 2(p− 1)/3 directions.