# Representations of torsion-free arithmetic matroids

## TL;DR

This paper introduces new methods to determine all representations of torsion-free arithmetic matroids, disproving existing conjectures about related posets through an algorithmic approach.

## Contribution

It develops a novel reduction operation and a signed Hermite normal form to compute all representations of torsion-free arithmetic matroids, advancing the understanding of their structure.

## Key findings

- Disproved two conjectures about the poset of layers and the independence poset.
- Provided an algorithm to enumerate all representations up to equivalence.
- Introduced new tools: reduction operation and signed Hermite normal form.

## Abstract

We study the representability problem for torsion-free arithmetic matroids. By using a new operation called "reduction" and a "signed Hermite normal form", we provide and implement an algorithm to compute all the representations, up to equivalence. As an application, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.