  Characteristic elements of the Tits algebra of a real hyperplane arrangement
carry information about the characteristic polynomial. We present this notion
and its basic properties, and apply it to derive various results about the
characteristic polynomial of an arrangement, from Zaslavsky's formulas to more
recent results of Kung and of Klivans and Swartz. We construct several examples
of characteristic elements, including one in terms of intrinsic volumes of
faces of the arrangement.
