Intersection problems for linear codes and polynomials over finite fields

This paper proves a stability result for a variation of the Erd\H{o}s-Ko-Rado theorem in the context of polynomials over finite fields. Let $\mathcal F$ be a family of polynomials of degree at most $k \geq 3$ in $\mathbb F_q[X]$. Call $\mathcal F$ intersecting if for any two polynomials $f, g$ in $\mathcal F$, there exists a point $x \in \mathbb F_q$ for which $f(x) = g(x)$. An intersecting family is called a star if it consists of all polynomials $f$ with ${\rm deg } f \leq k$ such that $f(x) = y$ for some fixed points $x, y \in \mathbb F_q$. In this paper we prove that if $\mathcal F$ is an intersecting family with $|\mathcal F| \geq \frac 1{\sqrt 2} q^k + \mathcal O(q^{k-1})$, then $\mathcal F$ is contained in a star. In fact, we prove that this is still true if we also evaluate the polynomials "at infinity", which is equivalent to studying the problem for homogeneous bivariate polynomials. The proof technique extends to a general framework for intersection problems of linear codes $C$. One has to investigate the geometry of the projective system $\mathcal S$ associated to $C$. If the hyperplanes that don't intersect $\mathcal S$ are well spread out with respect to the points not on $\mathcal S$, then one obtains stability results, showing that any intersecting family of reasonably large size is contained in a star.