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Let $V$ be a complex algebraic hypersurface defined by a polynomial $f$ withNewton polytope $\Delta$. It is well known that the spine of its amoeba has astructure of a tropical hypersurface. We prove in this paper that there existsa complex tropical hypersurface $V_{\infty, f}$ such that its coamoeba ishomeomorphic to the closure in the real torus of the coamoeba of $V$. Moreover,the coamoeba of $V_{\infty, f}$ contains an arrangement of $(n-1)$-torusdepending only on the geometry of $\Delta$ and the coefficients of $f$. Inaddition, we can consider this arrangement, as a weighted codual hyperplanesarrangement in the universal covering of the real torus, and the balancingcondition (the analogous to that of tropical hypersurfaces) is satisfied. Thiscodual hyperplanes arrangement is called the {\em shell} of the complexcoamoeba (the cousin of the spine of the complex amoeba). %(or the {\em averagecontour} of the complex coamoeba). Using this combinatorial coamoebasstructure, we show that the amoebas of complex algebraic hypersurfaces definedby maximally sparse polynomials are solid. More precisely, we characterize theimage of the order map defined by Forsberg, Passare, and Tsikh.

Geometric and Combinatorial Structure of Hypersurface Coamoebas