Amoebas and Coamoebas of Linear Spaces

Abstract

We give a complete description of amoebas and coamoebas of k-dimensional very affine linear spaces in (ℂ*)ⁿ. This include an upper bound of their dimension, and we show that if a k-dimensional very affine linear space in (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 π^2k. In addition, if the space is generic and real, then the volume of its amoeba is equal to π^2k/2^k.