On simplicial arrangements in $\mathbb{P}^3(\mathbb{R})$ with splitting polynomial

TL;DR

This paper investigates simplicial hyperplane arrangements in real projective 3-space, establishing conditions for their characteristic polynomials to have only real roots, leading to finiteness results and an updated classification catalog.

Contribution

It provides a necessary condition for the reality of roots of the characteristic polynomial and proves finiteness of certain simplicial arrangements with splitting polynomials.

Findings

Finiteness of isomorphism classes of simply laced simplicial arrangements with splitting polynomial

New combinatorial inequalities for arrangements with splitting polynomial

Updated catalogue of arrangements and review of conjectures

Abstract

In this paper, we study simplicial hyperplane arrangements in real projective $3$-space. We give a necessary condition for the characteristic polynomial to have only real roots, valid also for non-simplicial arrangements. As application, we obtain combinatorial inequalities which are satisfied for arrangements with splitting polynomial. This allows us to prove that there are only finitely many different isomorphism classes of simply laced simplicial arrangements whose characteristic polynomials split over $\mathbb{R}$. We also provide an updated version of a catalogue published by Gr\"unbaum and Shephard and review some conjectures of theirs.