Computing the Varchenko Determinant of a Bilinear Form

TL;DR

This paper presents a method to compute the exact Varchenko determinant for specific hyperplane arrangements by identifying relevant edges, advancing understanding of bilinear forms in geometric configurations.

Contribution

It introduces a way to determine the Varchenko determinant explicitly for certain arrangements using the concept of relevant edges.

Findings

Exact determinant values for specific arrangements are obtained.

Relevance of edges is key to computing the determinant.

Method simplifies calculations for complex arrangements.

Abstract

The Varchenko determinant is the determinant of the bilinear form associated to a real hyperplane arrangement. We show that we can obtain the exact value of this determinant for certain hyperplane arrangements if we know the edges which are relevant.