Non-Archimedean Coamoebae

Abstract

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.

Lagrangian submanifolds from tropical hypersurfaces

“…This is similar to other piecewise linear objects associated to tropical subvarieties in the context of complex geometry, e.g. the complexified non-archimedean amoeba in [12], also called phase tropical hypersurfaces (see for instance [9] or [14]).…”

Euler-Mellin integrals and A-hypergeometric functions

“…where f Γ is the truncated polynomial with support Γ. This has been proven by Johansson [Joh10] and independently by Nisse and Sottile [NS11].…”

Amoebas and Coamoebas of Linear Spaces

“…More precisely, amoebas degenerate to piecewise-linear objects called tropical varieties (see [13], and [19]), and comoebas degenerate to a non-Archimedean coamoebas which are the coamoebas of some lifting in the complex algebraic torus of tropical varieties. See [18] for more details about non-Archimedean coamoebas, and [16] about this degeneration in case of hypersurfaces. Whereas the theory of (co)amoebas of complex hypersurfaces is by now reasonably well understood (see e.g., [2], [11], [16], and [19],), much less is known about the structure of (co)amoebas coming from varieties of higher codimension.…”