Affine and toric hyperplane arrangements

TL;DR

This paper extends the mathematical framework connecting intersection and face lattices from central hyperplane arrangements to affine and toric cases, and generalizes Zaslavsky's results on counting regions on the torus.

Contribution

It introduces new mappings for affine and toric arrangements and generalizes fundamental counting results for regions on the torus.

Findings

Extended the Billera-Ehrenborg-Readdy map to affine and toric arrangements

Generalized Zaslavsky's results for counting regions on the torus

Provided new combinatorial tools for hyperplane arrangement analysis

Abstract

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.