Curve valuations and mixed volumes in the implicitization of rational varieties

We address the description of tropicalization families rational varieties under parametrizations with prescribed support, via curve valuations. recover and extend results by Sturmfels, Tevelev Yu for generic coefficients, considering non-trivial denominator. The advantage our point view is that it can be generalized to deal non-generic parametrizations. provide a detailed analysis degree closed image, based on combinatorial conditions relative positions supports polynomials defining parametrization. obtain new formula finer bounds degree, when are different. also present order at origin in case image hypersurface.