On computation of the characteristic polynomials of the discriminantal arrangements and the arrangements generated by generic points

TL;DR

This paper computationally explores the complex combinatorics of discriminantal arrangements, focusing on their characteristic polynomials and intersection lattices, especially for arrangements generated by generic points.

Contribution

It provides new computational results on the characteristic polynomials of discriminantal arrangements and reviews their intersection lattice structures.

Findings

Computed characteristic polynomials for specific discriminantal arrangements

Identified combinatorial properties of arrangements generated by generic points

Extended understanding of arrangements beyond Boolean and braid cases

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.