  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 Rees
and Sharp.
