Amoebas and their Tropicalizations – a Survey

Let V(f) be the complex hypersurface of a Laurent polynomial f. The amoeba A(f) is the projection of V(f) under the Log-absolute map. Amoebas have countless applications and, in particular, they form a key connection between “classical” algebraic geometry and tropical geometry. There exist multiple different tropical hypersurfaces related to amoebas. In this survey, we introduce the most important of these tropical hypersurfaces and compare their relations to amoebas. Moreover, we discuss related open problems in amoeba theory.