Amoeba

The amoeba of an algebraic variety is its image under the map

Here denotes the componentwise log-absolute-value map, not the complex logarithm. Thus is a subset of . The fiber over is the -torus

Consequently, forgets the complex argument of each coordinate and retains only the logarithm of its complex modulus. The corresponding image under the coordinatewise argument map is the coamoeba of . For a Laurent polynomial , the connected components of the complement set of the amoeba of are convex. Amoebas connect the geometry of complex varieties with convex geometry and tropical geometry. The mathematical literature generally uses the plural "amoebas" (e.g., Forsberg et al. 2000), while "amoebae" is common for the biological organisms.