Multiple addition, deletion and restriction theorems for hyperplane arrangements

TL;DR

This paper introduces new theorems for hyperplane arrangements, including deletion and restriction versions of the multiple addition theorem, enhancing understanding of their combinatorial properties and applications.

Contribution

It provides the first deletion version of the multiple addition theorem and generalizes existing theorems, advancing the theoretical framework of hyperplane arrangements.

Findings

Established the multiple deletion theorem (MDT)

Generalized the multiple addition theorem (MAT)

Applied results to extended Catalan arrangements

Abstract

In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT from the viewpoint of our new proof. Moreover, we introduce their restriction version, a multiple restriction theorem (MRT). Applications of them including the combinatorial freeness of the extended Catalan arrangements are given.