Combinatorics of line arrangements and dynamics of polynomial vector fields

TL;DR

This paper explores the relationship between line arrangements and polynomial vector fields, providing a dynamical perspective on derivations and showing that certain minimal degrees are not solely determined by combinatorial data.

Contribution

It offers a dynamical interpretation of derivation modules and demonstrates that the minimal degree of vector fields fixing finite line sets is not determined by combinatorics.

Findings

Dynamical interpretation of derivation modules as invariant polynomial vector fields.

Characterization of polynomial vector fields with infinitely many invariant lines.

Minimal degree of vector fields fixing finite lines is not combinatorially determined.

Abstract

Let $\mathcal{A}$ be a real line arrangement and $\mathcal{D}(\mathcal{A})$ the module of $\mathcal{A}$--derivations. First, we give a dynamical interpretation of $\mathcal{D}(\mathcal{A})$ as the set of polynomial vector fields which posses $\mathcal{A}$ as invariant set. We 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 $\mathcal{A}$.