# On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture

## TL;DR

This paper presents a parallel algorithm for generating all non-isomorphic rank 3 simple matroids with specific properties, applies it to analyze hyperplane arrangements, and provides evidence supporting Terao's freeness conjecture in certain cases.

## Contribution

The authors develop a parallel algorithm for generating rank 3 simple matroids and create a comprehensive database, providing new insights into Terao's freeness conjecture.

## Key findings

- Smallest divisionally free rank 3 arrangement has 14 hyperplanes in all characteristics except 2 and 5.
- Terao's freeness conjecture holds for rank 3 arrangements with 14 hyperplanes in any characteristic.
- Generated and stored all such matroids with various invariants in an accessible database.

## Abstract

In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics distinct from $2$ and $5$. Another database query proves that Terao's freeness conjecture is true for rank $3$ arrangements with $14$ hyperplanes in any characteristic.