  We prove a formula for the multidegrees of a rational map defined by
generalized monomials on a projective variety, in terms of integrals over an
associated Newton region. This formula leads to an expression of the
multidegrees as volumes of related polytopes, in the spirit of the classical
Bernstein-Kouchnirenko theorem, but extending the scope of these formulas to
more general monomial maps. We also determine a condition under which the
multidegrees may be computed in terms of the characteristic polynomial of an
associated matrix.
