Amoeba-shaped polyhedral complex of an algebraic hypersurface

TL;DR

This paper introduces a new polyhedral complex related to algebraic hypersurfaces that captures key topological and combinatorial properties of amoebas, providing explicit formulas and descriptions especially for optimal cases.

Contribution

It defines a novel amoeba-inspired polyhedral complex within the Newton polytope and offers explicit formulas for cases with dual triangulations and optimal hypersurfaces.

Findings

Provides an explicit formula for the polyhedral complex.

Describes the complex for hypersurfaces with dual triangulations.

Characterizes the complex for optimal hypersurfaces.

Abstract

Given a complex algebraic hypersurface~$H$, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~$H$ and enjoys the key topological and combinatorial properties of the amoeba of~$H.$ We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal.