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

Given a hyperplane arrangement A of R n whose defining equations have integer coefficients, the reduction of A modulo q gives rise to a group arrangement A q of ( Z / q Z ) n . We study the restriction A B of A to a subspace B x = 0 of R n with B an integral matrix, and its reduction A q B modulo q. We show that the counting function F ( A B , q ) of the number of elements of the complement of A q B is a quasi-polynomial function of q, and can be written in the form F ( A B , q ) = ∑ j = r s ( − 1 ) j β j ( q ) q n − j . If a, b are positive integers and a divides b, then β j ( b ) ⩾ β j ( a ) ⩾ 0 . In particular, if A B is a hyperplane arrangement, we have β j ( q ) ⩾ b j , where b j are the absolute values of the coefficients of the characteristic polynomial χ ( A B , t ) .