Higher convexity of coamoeba complements

TL;DR

This paper proves that the complement of the coamoeba of certain algebraic varieties exhibits k-convexity, extending previous results and applying to nonarchimedean cases, advancing understanding of geometric properties of coamoebas.

Contribution

It generalizes the convexity property of coamoeba complements from hypersurfaces to higher codimension varieties and their nonarchimedean counterparts.

Findings

Complement of coamoeba of codimension k+1 variety is k-convex.

Extends Nisse's result from hypersurfaces to higher codimension.

Nonarchimedean coamoeba complements are also k-convex.

Abstract

We show that the complement of the coamoeba of a variety of codimension k+1 is k-convex, in the sense of Gromov and Henriques. This generalizes a result of Nisse for hypersurface coamoebas. We use this to show that the complement of the nonarchimedean coamoeba of a variety of codimension k+1 is k-convex.