Contour of an amoeba

For $f \in \mathbb{C}[x_1,x_2]$, let $\mathcal{C}_f \subset \mathbb{R}^2$ denote the contour of the amoeba of $f$, i.e., the locus of the critical points of the Gauss map. The singular points $V$ on $\mathcal{C}_f$ naturally divides $\mathcal{C}_f$ into several arcs $E$, and thus $(V,E)$ defines a planar graph.

Question: What combinatorial properties does the graph $(V,E)$ have? Which graphs can be realized by some function $f$?

Background: Some examples of the contour can be found e.g., in T. Theobald, Computing amoebas, Exp. Math. 11:513-526, 2002, or in M. Passare and A. Tsikh, Amoebas: their spines and their contours, Preprint, 2003. Since tracing the contour can be used to (numerically) compute the boundary of the amoeba,understanding the combinatorial properties of the contour helps to compute the boundary of the amoeba.

Question: How does this generalize to higher dimension?

(contributed by Thorsten Theobald)

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