Jens Forsgård: Lopsided Coamoebas

The coamoeba of an algebraic hypersurface V(f) ⊂  (ℂ_*)^n is defined as its image under the componentswise argument mapping. One of the main problems regarding coamoebas is to describe the structure of its set of connected complement components, given either a fixed point configuration A = supp(f) or a fixed Newton polytope Δ_f = Conv(A). The progress so far consists of an upper bound given by the weightened volume n! Vol(Δ_f). In this talk we will introduce the lopsided coamoeba. This is a simpler object then the coamoeba, and in general it has fewer complement components. We will show that there is a relation between its complement components and a certain translated lattice inside the zonotope of a Gale transform of A. In some cases this allows us to construct coamoebas with the maximal number of complement components. Focusing on the exceptional cases, we find some interesting examples of coamoebas.