Tutte polynomials of hyperplane arrangements and the finite field method

TL;DR

This survey explores the Tutte polynomial's role in hyperplane arrangements, illustrating its connections to various invariants and demonstrating a finite field method for computation.

Contribution

It highlights the relationships between Tutte polynomials and invariants of hyperplane arrangements and introduces a finite field method for computing Tutte polynomials.

Findings

Many invariants can be expressed via the Tutte polynomial

Hyperplane arrangements provide a finite field method for computation

The Tutte polynomial links combinatorial, algebraic, geometric, and topological properties

Abstract

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements. We show that many enumerative, algebraic, geometric, and topological invariants of a hyperplane arrangement can be expressed in terms of its Tutte polynomial. We also show that, even if one is only interested in computing the Tutte polynomial of a graph or a matroid, the theory of hyperplane arrangements provides a powerful finite field method for this computation.