Amoebas and their Tropicalizations – a Survey

Abstract

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.

As a new contribution we provide an example of an amoeba in ℝ² which has a component in the complement with an order not contained in the support of the defining polynomial. As a consequence, we conclude that an amoeba and its corresponding complement induced tropical hypersurface are not homotopy equivalent in general. Similarly, we prove that Archimedean amoebas and non-Archimedean amoebas are not homotopy equivalent in general.