Coamoebas of complex algebraic plane curves and the logarithmic Gauss map

The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\mathbb{C}^*)^2$, the complex algebraic plane curves whose coamoebas are of maximal area (counted with multiplicity) are defined over $\mathbb{R}$, and their real loci are Harnack curves possibly with ordinary real isolated double points (c.f. \cite{MR-00}). In addition, we characterize the complex algebraic plane curves such that their coamoebas contain no extra-piece.