Discriminant coamoebas through homology

TL;DR

This paper provides a new topological proof for the structure of coamoebas and their complements related to A-discriminants, extending previous results from two dimensions to all dimensions.

Contribution

It introduces a topological approach to analyze coamoebas of A-discriminants, generalizing prior two-dimensional results to higher dimensions.

Findings

Reproves the coamoeba structure in dimension two using topology

Extends the coamoeba and zonotope cycle result to all dimensions

Provides a new, more general proof method for these structures

Abstract

Understanding the complement of the coamoeba of a (reduced) A-discriminant is one approach to studying the monodromy of solutions to the corresponding system of A-hypergeometric differential equations. Nilsson and Passare described the structure of the coamoeba and its complement (a zonotope) when the reduced A-discriminant is a function of two variables. Their main result was that the coamoeba and zonotope form a cycle which is equal to the fundamental cycle of the torus, multiplied by the normalized volume of the set A of integer vectors. That proof only worked in dimension two. Here, we use simple ideas from topology to give a new proof of this result in dimension two, one which can be generalized to all dimensions.