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Division of characteristic polynomials and the division theorem for free arrangements of hyperplanes

We consider the triple (A,A′,AH) of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of A and the division of χ(A; t) by χ(A ; t) confirm the freeness ofA. The key ingredient of this “division theorem” on freeness is the fact that, if χ(A ; t) divides χ(A; t), then the same holds for the localization at the codimension three flat in H. This implies the local-freeness of A in codimension three along H. Based on these results, several applications are obtained, which include a definition of “divisionally free arrangements”. It is strictly larger than the set of inductively free arrangements. Also, in the set of divisionally free arrangements, the Terao’s conjecture is true.