Combinatorics of line arrangements and polynomial vector fields

TL;DR

This paper studies the properties of polynomial vector fields related to real line arrangements, characterizing those with infinitely many invariant lines and showing that the minimal degree for fixing only finitely many lines is not determined by combinatorial data.

Contribution

It characterizes polynomial vector fields with infinitely many invariant lines and demonstrates that the minimal degree fixing only finitely many lines is not solely determined by combinatorics.

Findings

Polynomial vector fields with infinitely many invariant lines are characterized.

The minimal degree of polynomial vector fields fixing only finitely many lines is not determined by combinatorics.

The relationship between line arrangements and polynomial vector fields is clarified.

Abstract

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$-derivations view as the set of polynomial vector fields which possess $\mathcal{A}$ as an invariant set. We first characterize polynomial vector fields having an infinite number of invariant lines. Then we prove that the minimal degree of polynomial vector fields fixing only a finite set of lines in $\mathcal{D}(\mathcal{A})$ is not determined by the combinatorics of $\mathcal{A}$.