Mixed multiplicities of filtrations

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of m_R-primary ideals in a Noetherian local ring R, generalizing the classical theory for m_R-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R, which holds, for instance, if R is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of m_R-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and of Rees and Sharp.