# Generalized threshold arrangements

## TL;DR

This paper develops a method to compute the characteristic polynomial of generalized threshold hyperplane arrangements, extending previous results on specific arrangements like Shi and Catalan.

## Contribution

It introduces a new approach for calculating characteristic polynomials of a broad class of threshold arrangements, generalizing prior specific cases.

## Key findings

- Derived a formula for the characteristic polynomial of generalized threshold arrangements
- Extended known results from Shi and Catalan arrangements to a broader class
- Provided a systematic method applicable to various threshold arrangements

## Abstract

An arrangement of hyperplanes is a finite collection of hyperplanes in a real Euclidean space. To such a collection one associates the characteristic polynomial that encodes the combinatorics of intersections of the hyperplanes. Finding the characteristic polynomial of the Shi threshold and the Catalan threshold arrangements was an open problem in Stanley's list of problems in [1]. Seunghyun Seo solved both the problems by clever arguments using the finite field method in [3,4]. However, in his paper, he left open the problem of computing the characteristic polynomial of a broader class of threshold arrangements, the so-called "generalized threshold" arrangements whose defining set of hyperplanes is given by $x_i + x_j = -l,-l+1,...,m-1,m$ for $1 \le i < j \le n$ where $l,m \in \mathbb{N}$. In this paper, we present a method for computing the characteristic polynomial of this family of arrangements.