Intersecting families of graphs of functions over a finite field

TL;DR

This paper investigates intersecting families of polynomial graphs over finite fields, establishing size bounds, stability results, and characterizations of graphs with specific directional properties, using polynomial and combinatorial methods.

Contribution

It provides a stability version of a classical bound on intersecting polynomial families and characterizes graphs with restricted directions over finite fields.

Findings

Maximum size of intersecting polynomial families is $q^k$.

Stability holds for sizes greater than $q^k - q^{k-1}$.

Graphs with direction sets in additive subgroups are lines.

Abstract

Let $U$ be a set of polynomials of degree at most $k$ over $\mathbb{F}_q$, the finite field of $q$ elements. Assume that $U$ is an intersecting family, that is, the graphs of any two of the polynomials in $U$ share a common point. Adriaensen proved that the size of $U$ is at most $q^k$ with equality if and only if $U$ is the set of all polynomials of degree at most $k$ passing through a common point. In this manuscript, using a different, polynomial approach, we prove a stability version of this result, that is, the same conclusion holds if $|U|>q^k-q^{k-1}$. We prove a stronger result when $k=2$. For our purposes, we also prove the following results. If the set of directions determined by the graph of $f$ is contained in an additive subgroup of $\mathbb{F}_q$, then the graph of $f$ is a line. If the set of directions determined by at least $q-\sqrt{q}/2$ affine points is contained in the set of squares/non-squares plus the common point of either the vertical or the horizontal lines, then up to an affinity the point set is contained in the graph of some polynomial of the form $\alpha x^{p^k}$.