On the Smith normal form of Varchenko matrix

Abstract

Let A be a hyperplane arrangement in A isomorphic to ℝ n . Let V q be the q -Varchenko matrix for the arrangement A with all hyperplane parameters equal to q . In this paper, we consider three interesting cases of q -Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over ℤ[ q ] .