Comparison on the coefficients of characteristic quasi-polynomials of integral arrangements

Given a hyperplane arrangement AA of RnRn whose defining equations have integer coefficients, the reduction of AA modulo q   gives rise to a group arrangement AqAq of n(Z/qZ)(Z/qZ)n. We study the restriction ABAB of AA to a subspace Bx=0Bx=0 of RnRn with B   an integral matrix, and its reduction AqB modulo q  . We show that the counting function F(AB,q)F(AB,q) of the number of elements of the complement of AqB is a quasi-polynomial function of q, and can be written in the formF(AB,q)=∑j=rs(−1)jβj(q)qn−j. If a, b are positive integers and a divides b  , then βj(b)⩾βj(a)⩾0βj(b)⩾βj(a)⩾0. In particular, if ABAB is a hyperplane arrangement, we have βj(q)⩾bjβj(q)⩾bj, where bjbj are the absolute values of the coefficients of the characteristic polynomial χ(AB,t)χ(AB,t).