Coamoebas of complex algebraic plane curves and the logarithmic Gauss map

TL;DR

This paper investigates the properties of coamoebas of complex algebraic plane curves, establishing bounds on their area, characterizing those with maximal coamoeba area as real Harnack curves, and describing conditions for coamoebas with no extra pieces.

Contribution

It characterizes complex algebraic plane curves with maximal coamoeba area as real Harnack curves and describes conditions for coamoebas to have no extra pieces.

Findings

Maximal coamoeba area corresponds to real Harnack curves.

Curves with no extra-piece coamoebas are characterized.

Area bounds are related to the degree of the curve.

Abstract

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.