Complex tropical localization, coamoebas, and mirror tropical hypersurfaces

TL;DR

This paper introduces tropical mirror hypersurfaces, proves a complex tropical localization theorem extending Kapranov's theorem, and establishes geometric and topological equivalences between coamoebas of complex and tropical hypersurfaces.

Contribution

It presents the concept of tropical mirror hypersurfaces and a new localization theorem in tropical geometry, linking complex algebraic and tropical hypersurfaces.

Findings

Proves a complex tropical localization theorem extending Kapranov's theorem.

Establishes geometric equivalence between coamoebas of complex and tropical hypersurfaces.

Shows topological equivalence between coamoebas of maximally sparse complex and tropical hypersurfaces.

Abstract

We introduce in this paper the concept of tropical mirror hypersurfaces and we prove a complex tropical localization Theorem which is a version of Kapranov's Theorem \cite{K-00} in tropical geometry. We give a geometric and a topological equivalence between coamoebas of complex algebraic hypersurfaces defined by a maximally sparse polynomial and coamoebas of maximally sparse complex tropical hypersurfaces.