  We consider the notion of mixed multiplicities for multigraded modules by
using Hilbert series, and this is later applied to study the projective degrees
of rational maps. We use a general framework to determine the projective
degrees of a rational map via a computation of the multiplicity of the
saturated special fiber ring. As specific applications, we provide explicit
formulas for all the projective degrees of rational maps determined by perfect
ideals of height two or by Gorenstein ideals of height three.
