  We develop a theory of multiplicities and mixed multiplicities of
filtrations, extending the theory for filtrations of $m$-primary ideals to
arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of
$r$ filtrations on an analytically unramified local ring $R$ come from the
coefficients of a suitable homogeneous polynomial in $r$ variables of degree
equal to the dimension of the ring, analogously to the classical case of the
mixed multiplicities of $m$-primary ideals in a local ring. We prove that the
Minkowski inequalities hold for arbitrary filtrations. The characterization of
equality in the Minkowski inequality for m-primary ideals in a local ring by
Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but
we show that they are true in a large and important subcategory of filtrations.
We define divisorial and bounded filtrations. The filtration of powers of a
fixed ideal is a bounded filtration, as is a divisorial filtration. We show
that in an excellent local domain, the characterization of equality in the
Minkowski equality is characterized by the condition that the integral closures
of suitable Rees like algebras are the same, strictly generalizing the theorem
of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees
characterizing the inclusion of ideals with the same multiplicity generalizes
to bounded filtrations in excellent local domains. We give a number of other
applications, extending classical theorems for ideals.
