A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. We describe the structure of the boundary of the coamoeba of a variety, which we relate to its logarithmic limit set. Detailed examples of lines in three-dimensional space illustrate and motivate these results.

Abstract. 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 give 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 any higher order iterative polynomial system solving algorithm, such as homotopy continuation. 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.In memory of Mikael Passare.

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Abstract. A coamoeba is the image of a subvariety of a complex torus under the argument map to the real torus. Similarly, a non-archimedean coamoeba is the image of a subvariety of a torus over a non-archimedean field K with complex residue field under an argument map. The phase tropical variety is the closure of the image under the pair of maps, tropicalization and argument. We describe the structure of non-archimedean coamoebae and phase tropical varieties in terms of complex coamoebae and their phase limit sets. The argument map depends upon a section of the valuation map, and we explain how this choice (mildly) affects the non-archimedean coamoeba. We also identify a class of varieties whose non-archimedean coamoebae and phase tropical varieties are objects from polyhedral combinatorics.

Abstract. We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.