Intersecting families and nonvanishing multivariate polynomials over finite fields

Abstract

Let $\mathcal{P}_{n,d}$ be the space of polynomials in $n$ variables over $\mathbb{F}_q$ of degree at most $d$. Two polynomials $f,g\in\mathcal{P}_{n,d}$ intersect if $f(\mathbf a)=g(\mathbf a)$ for some $\mathbf a\in\mathbb{F}_q^n$. A star consists of all polynomials $f\in\mathcal{P}_{n,d}$ satisfying $f(\mathbf a)=b$ for fixed $\mathbf a\in\mathbb{F}_q^n$ and $b\in\mathbb{F}_q$. We completely classify the maximum intersecting families in $\mathcal{P}_{n,d}$. When $n=1$ and $d\geq 2$, it was previously shown that all maximum intersecting families are stars. We prove that the same conclusion holds for all $n\geq 2$ and $d\geq 2$ when $q$ is odd. When $q$ is even, however, the situation is more subtle, and a new phenomenon emerges: for $q\geq 4$, maximum non-star examples exist precisely when $d\leq n$. Along the way, we prove two further results of independent interest. First, we determine the span of nonvanishing polynomials in $\mathcal{P}_{n,d}$. Second, we characterize all linear functionals $Ψ\colon\mathcal{P}_{n,d}\to\mathbb{F}_q$ whose kernels are disjoint from the set of nonvanishing polynomials. The first result plays a crucial role in the proof of our main result; the second is a Gleason--Kahane--Żelazko theorem for polynomials of bounded degree over finite fields.

Disclosure

“Karen Meagher for helpful discussions. The research of the third author was supported in part by an NSERC fellowship. AI DISCLOSURE During the preparation of this paper, the authors used AI tools interactively as an aid in dis- cussing, refining, and presenting parts of the manuscript. The mathematical ideas, underlying intuition, and overall direction are due to the authors. The authors are solely responsible for the final content”

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2026-08-23T05:22:40.592256Z

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