  Let $V$ be a complex algebraic hypersurface defined by a polynomial $f$ with
Newton polytope $\Delta$. It is well known that the spine of its amoeba has a
structure of a tropical hypersurface. We prove in this paper that there exists
a complex tropical hypersurface $V_{\infty, f}$ such that its coamoeba is
homeomorphic to the closure in the real torus of the coamoeba of $V$. Moreover,
the coamoeba of $V_{\infty, f}$ contains an arrangement of $(n-1)$-torus
depending only on the geometry of $\Delta$ and the coefficients of $f$. In
addition, we can consider this arrangement, as a weighted codual hyperplanes
arrangement in the universal covering of the real torus, and the balancing
condition (the analogous to that of tropical hypersurfaces) is satisfied. This
codual hyperplanes arrangement is called the {\em shell} of the complex
coamoeba (the cousin of the spine of the complex amoeba). %(or the {\em average
contour} of the complex coamoeba). Using this combinatorial coamoebas
structure, we show that the amoebas of complex algebraic hypersurfaces defined
by maximally sparse polynomials are solid. More precisely, we characterize the
image of the order map defined by Forsberg, Passare, and Tsikh.

Algebraic Geometry;Complex Variables;Geometric Topology