The Euler multiplicity and addition–deletion theorems for multiarrangements

By Takuro Abe, Hiroaki Terao, Max Wakefield. 2006

57 citations recorded by the source corpus.

The addition–deletion theorems for hyperplane arrangements, which were originally shown by Terao [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) 293–320.], provide useful ways to construct examples of free arrangements. In this article, we prove addition–deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition–deletion theorems to various arrangements, including supersolvable arrangements and the Coxeter arrangement of type A3, to construct free and non‐free multiarrangements.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/bf8a3f3426728f42339ddc61d6835ed542add2b2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. H. Schenck
2. Takuro Abe
3. Alexandru Dimca

In Discrete Mathematics and Combinatorics

1. H. Schenck
2. Takuro Abe
3. Stefan O. Tohaneanu

In Electrical and Electronic Engineering

1. Takuro Abe
2. Stefan O. Tohaneanu

Citations in the corpus

• Freeness of conic-line arrangements in P2 (2009, result): “An addition–deletion theorem for multiarrangements has recently been proven by Abe–Terao–Wakefield in [1]; our results are the first (to our knowledge) to give an inductive criterion for freeness for nonlinear arrangements.”
• Divisionally free arrangements of hyperplanes (2015, methodology): “Also, let us recall the addition-deletion theorem for multiarrangements (A, m) from [8].”