Tutte polynomials of hyperplane arrangements and the finite field method

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement, which answers a wide variety of questions about its underlying object. 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 easily 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. We begin by discussing the basic definitions on hyperplane arrangements and their characteristic and Tutte polynomials in Sections 1 and 2, respectively. Section 3 discusses numerous applications of Tutte polynomials of hyperplane arrangements in combinatorics, algebra, and geometry. Section 4 discusses arrangements over finite fields, and the Finite Field Method for computing Tutte polynomials. Finally, in Section 5 we collect the most interesting arrangements whose characteristic and Tutte polynomials are known. Our presentation is heavily influenced by a 2002 graduate course on Hyperplane Arrangements by Richard Stanley at MIT, much of which became part of [Sta07]. See [OT92] for a great introduction to more algebraic and topological aspects of the theory of hyperplane arrangements.