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Many combinatorial and topological invariants of a hyperplane arrangement canbe computed in terms of its Tutte polynomial. Similarly, many invariants of ahypertoric arrangement can be computed in terms of its arithmetic Tuttepolynomial. We compute the arithmetic Tutte polynomials of the classical rootsystems of types A,B,C, and D, with respect to their integer, root, and weightlattices. We do it in two ways: by introducing a finite field method forarithmetic Tutte polynomials, and by enumerating signed graphs with respect tosix parameters.

The arithmetic Tutte polynomials of the classical root systems