Discriminant coamoebas in dimension two

TL;DR

This paper characterizes the coamoebas of plane algebraic curves defined by A-discriminants, providing explicit descriptions, area formulas, and their relation to zonotopes, revealing topological and geometric properties.

Contribution

It offers a novel explicit description of coamoebas of A-discriminant curves in two variables, linking them to polygons and zonotopes, and establishes their topological and measure-theoretic properties.

Findings

Coamoebas are unions of two mirror images of a polygon.

The area of the coamoeba can be explicitly computed.

Coamoebas and zonotopes together form a cycle covering the torus multiple times.

Abstract

This paper deals with coamoebas, that is, images under coordinatewise argument mappings, of certain quite particular plane algebraic curves. These curves are the zero sets of reduced A-discriminants of two variables. We consider the coamoeba primarily as a subset of the torus T^2=(R/2\pi Z)^2, but also as a subset of its covering space R^2, in which case the coamoeba consists of an infinite, doubly periodic image. In fact, it turns out to be natural to take multiplicities into account, and thus to treat the coamoeba as a chain in the sense of algebraic topology. We give a very explicit description of the coamoeba as the union of two mirror images of a (generally non-convex) polygon, which is easily constructed from a matrix B that represents the Gale transform of the original collection A. We also give an area formula for the coamoeba, and we show that the coamoeba is intimately related to a certain zonotope. In fact, on the torus T^2 the coamoeba and the zonotope together form a cycle, and hence precisely cover the entire torus an integer number of times. This integer is proved to be equal to the (normalized) volume of the convex hull of A.