Non-Archimedean coamoebae

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 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 phaseropical 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.