Coamoebas of complex algebraic plane curves and the logarithmic Gauss map

Details Coamoebas of complex algebraic plane curves and the logarithmic Gauss map Coamoebas of complex algebraic plane curves and the logarithmic Gauss map

2008-05-19

arxiv.org/abs/0805.2872v3

Mathematics

Algebraic Geometry

11 pages, 2 figure

Scientific paper

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.

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