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

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, for example, many integer-rooted non-free arrangements have been found in [9]. However, still main examples whose characteristic polynomials factor over the integer ring are free arrangements. 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.