Amoeba

Summary

The amoeba of a complex polynomial in $n$ variables is the image of its zero set in $(\C^\times)^n$ under a logarithmic map. This map takes the logarithm of the absolute value of each coordinate. The name, given by Gelfand, Kapranov and Zelevinsky, comes from its shape: a body with thin tentacles going off to infinity. Each connected component of the complement is convex, and the number of these components is bounded by the Newton polytope. Amoebas link complex algebraic geometry with convex geometry and degenerate to tropical varieties, see tropical geometry.