Amoebas and coamoebas of linear spaces

TL;DR

This paper provides a comprehensive description of amoebas and coamoebas of linear spaces in complex tori, including their dimensions and volumes, with results for generic and real cases.

Contribution

It offers a complete characterization of amoebas and coamoebas of linear spaces, including bounds on their dimensions and explicit volume formulas for generic and real cases.

Findings

Dimension of (co)amoebas for generic spaces is min{2k, n}.

Volume of coamoeba equals π^{2k}.

Volume of amoeba for real generic spaces is π^{2k}/2^k.

Abstract

We give a complete description of amoebas and coamoebas of $k$-dimensional very affine linear spaces in $(\mathbb{C}^*)^{n}$. This include an upper bound of their dimension, and we show that if a $k$-dimensional very affine linear space in $(\mathbb{C}^*)^{n}$ is generic, then the dimension of its (co)amoeba is equal to $\min \{ 2k, n\}$. Moreover, we prove that the volume of its coamoeba is equal to $\pi^{2k}$. In addition, if the space is generic and real, then the volume of its amoeba is equal to $\frac{\pi^{2k}}{2^k}$.