Metric estimates and membership complexity for Archimedean amoebae and tropical hypersurfaces

Given any complex Laurent polynomial f, Amoeba(f) is the image of its complex zero set under the coordinate-wise log absolute value map. We discuss an efficiently constructible polyhedral approximation, ArchTrop(f), of Amoeba(f), and derive explicit upper and lower bounds, solely as a function of the number of monomial terms of f, for the Hausdorff distance between these two sets. We also show that deciding whether a given point lies in ArchTrop(f) is doable in polynomial-time, for any fixed dimension, unlike the corresponding problem for Amoeba(f), which is NP-hard already in one variable. ArchTrop(f) can thus serve as a canonical low order approximation to start a higher order iterative polynomial system solving algorithm, e.g., homotopy continuation.