The Tutte polynomial of ideal arrangements

The Tutte polynomial was originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. in 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for certain prime powers of the first variable at the same time. In this paper, we compute the Tutte polynomial of ideal arrangements. These arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root. systems. For the ideals of classical root systems, we bring a slight improvement of the finite field method by showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to the studied hyperplane arrangement.. Computing the minor set. associated to an ideal of classical root systems particularly permits us to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G(2), F-4, and E-6, we use the formula of Crapo.