On simplicial arrangements in ℙ³(ℝ) with splitting polynomial

2019/02/28 by Geis, David
#52C35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.11185

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 ℝ. We also provide an updated version of a catalogue published by Grünbaum and Shephard and review some conjectures of theirs.

Related

• Combinatorics of free and simplicial line arrangements
• Combinatorial simpliciality of arrangements of hyperplanes
• Simplicial arrangements with up to 27 lines
• Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations
• Topology and combinatorics of real line arrangements