  Given any complex Laurent polynomial $f$, $\mathrm{Amoeba}(f)$ is the image
of its complex zero set under the coordinate-wise log absolute value map. We
give an efficiently constructible polyhedral approximation,
$\mathrm{ArchtTrop}(f)$, of $\mathrm{Amoeba}(f)$, 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 $\mathrm{ArchTrop}(f)$ is doable in
polynomial-time, for any fixed dimension, unlike the corresponding problem for
$\mathrm{Amoeba}(f)$, which is $\mathbf{NP}$-hard already in one variable.
$\mathrm{ArchTrop}(f)$ can thus serve as a canonical low order approximation to
start any higher order iterative polynomial system solving algorithm, such as
homotopy continuation. $\mathrm{ArchTrop}(f)$ also provides an Archimedean
analogue of Kapranov's Non-Archimedean Amoeba Theorem and a higher-dimensional
extension of earlier estimates of Mikhalkin and Ostrowski.
