Intersecting families of polynomials over finite fields

Abstract

This paper demonstrates an analog of the Erdős–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 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 Erdős–Ko–Rado theorem, which establishes a sharp upper bound on the size of these families. Here, we extend the Erdős–Ko–Rado theorem to polynomial rings over finite fields.

Specifically, 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 prove that the upper bound for this size is $q^{n-\ell }$ and characterize all extremal families that achieve this maximum size.

Extending 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.