The Varchenko Matrix for Dehyperplane Arrangement

TL;DR

This paper extends the computation of the Varchenko determinant to dehyperplane arrangements on real manifolds, exploring their properties and solution spaces, which differ from traditional hyperplane arrangements.

Contribution

It introduces the calculation of the Varchenko determinant for dehyperplane arrangements and analyzes their solution spaces, broadening understanding beyond central hyperplane arrangements.

Findings

Computed the Varchenko determinant for dehyperplane arrangements.

Analyzed the solution space of associated linear systems.

Identified differences from classical hyperplane arrangements.

Abstract

This article computes the Varchenko determinant of dehyperplane arrangements which are generalizations of pseudohyperplane arrangements. But unlike those latter, they are defined on a real manifold, and it is not always possible to obtain a central dehyperplane arrangement by coning. This article also studies the solution space of a linear system defined from a dehyperplane arrangement. That equation system was first introduced by Aguiar and Mahajan for central hyperplane arrangements.