On Computation of the Characteristic Polynomials of the Discriminantal Arrangements and the Arrangements Generated by Generic Points

Abstract

In this article we give a computational study of combinatorics of the discriminantal arrangements. The discriminantal arrangements are parametrized by two positive integers n and k such that n>k. The intersection lattice of the discriminantal arrangement with the parameter (n,k) is isomorphic to the intersection lattice of the hyperplane arrangement generated by n generic points in the d-dimensional vector space where d=n-k-1. The combinatorics of the discriminantal arrangements is very hard, except for the special cases of the Boolean arrangements (k=0) and the braid arrangements (k=1). We review some results on the intersection lattices of the arrangements generated by generic points and use them to obtain some computational results on the characteristic polynomials of the discriminantal arrangements.