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

Abstract

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 > 2 p − 1 − 11 4 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 − 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 [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2 ( p − 1 ) / 3 directions.