Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations

TL;DR

This paper investigates the roots of characteristic polynomials of hyperplane arrangements, focusing on free arrangements, their restrictions, and localizations, and how these roots behave over integers and reals.

Contribution

It explores the behavior of roots of characteristic polynomials in free arrangements, especially under restrictions and localizations, extending Terao's factorization theorem.

Findings

Characteristic polynomial roots often remain integral or real in free arrangements.

Restrictions and localizations of free arrangements tend to preserve polynomial factorization.

Analysis of non-free arrangements with factorable characteristic polynomials.

Abstract

Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many non-free arrangements whose characteristic polynomial factors over the integer ring. On the other hand, the localization of a free arrangement is free, and its restriction is in many cases free, thus its characteristic polynomial factors. In this paper, we consider how their integer, or real roots behave.