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Given any complex Laurent polynomial $f$, $\mathrm{Amoeba}(f)$ is the imageof its complex zero set under the coordinate-wise log absolute value map. Wegive an efficiently constructible polyhedral approximation,$\mathrm{ArchtTrop}(f)$, of $\mathrm{Amoeba}(f)$, and derive explicit upper andlower bounds, solely as a function of the number of monomial terms of $f$, forthe Hausdorff distance between these two sets. We also show that decidingwhether a given point lies in $\mathrm{ArchTrop}(f)$ is doable inpolynomial-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 tostart any higher order iterative polynomial system solving algorithm, such ashomotopy continuation. $\mathrm{ArchTrop}(f)$ also provides an Archimedeananalogue of Kapranov's Non-Archimedean Amoeba Theorem and a higher-dimensionalextension of earlier estimates of Mikhalkin and Ostrowski.

Metric Estimates and Membership Complexity for Archimedean Amoebae and Tropical Hypersurfaces