  We give formulas and effective sharp bounds for the degree of multi-graded
rational maps and provide some effective and computable criteria for
birationality in terms of their algebraic and geometric properties. We also
extend the Jacobian dual criterion to the multi-graded setting. Our approach is
based on the study of blow-up algebras, including syzygies, of the ideal
generated by the defining polynomials of the rational map. A key ingredient is
a new algebra that we call the "saturated special fiber ring", which turns out
to be a fundamental tool to analyze the degree of a rational map. We also
provide a very effective birationality criterion and a complete description of
the equations of the associated Rees algebra of a particular class of plane
rational maps.
