Intersecting families of polynomials over finite fields

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.