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 ${\mathbb F}_p$ with a prime number $p$ of elements, to ${\mathbb F}_p$. In this article we consider those functions $f(X)$ for which there is a positive integer $n > 2\sqrt{p-1}-\frac{11}{4}$ with the property that $f(X)^i$, when considered as an element of ${\mathbb F}_p [X]/(X^p-X)$, has degree at most $p-2-n+i$, for all $i=1,\ldots,n$. 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 $p-2\sqrt{p-1}+\frac{11}{4}$ directions. In particular, the proof of the conjecture for $t=2$ and $t=3$ gives new proofs of the result of Lov\'asz and Schrijver \cite{LS1981} and the result in \cite{Gacs2003} respectively, which classify all functions which determine at most $2(p-1)/3$ directions.