Non-Archimedean coamoebae

TL;DR

This paper explores the structure of non-archimedean coamoebae and phase tropical varieties, linking them to complex coamoebae and polyhedral combinatorics, and discusses how valuation sections influence their shape.

Contribution

It provides a detailed description of non-archimedean coamoebae and phase tropical varieties, connecting them to complex coamoebae and analyzing the impact of valuation sections.

Findings

Non-archimedean coamoebae can be described using complex coamoebae.

Phase tropical varieties relate to polyhedral combinatorics.

The choice of valuation section mildly affects the coamoeba shape.

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.