Intersecting families of polynomials over finite fields

TL;DR

This paper proves an analog of the Erd ext{"o}s-Ko-Rado theorem for polynomial families over finite fields, determining maximum sizes of intersecting polynomial families and characterizing extremal cases.

Contribution

It establishes the maximum size of intersecting polynomial families over finite fields and characterizes all extremal families, extending classical combinatorial theorems to algebraic structures.

Findings

Maximum size of pairwise intersecting polynomial families is q^{n-l}.

Only trivial families achieve maximum size in triple-intersecting cases.

Trivial families are the only extremal solutions when polynomials have degree at most n.

Abstract

This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting if any two subsets in the family intersect in at least $l$ elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erd\H{o}s-Ko-Rado theorem, which establishes a sharp upper bound on the size of these families. As an analog of the Erd\H{o}s-Ko-Rado theorem, we determine the largest possible size of a family of monic polynomials, each of degree $n$, over a finite field $F_q$, where every pair of polynomials in the family shares a common factor of degree at least $l$. We establish that the upper bound for this size is $q^{n-l}$ and characterize all extremal families that achieve this maximum size. Further extending our study to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least $l$, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most $n$, we affirm that only trivial families achieve the corresponding upper bound.