Discriminant and Singularities of Logarithmic Gauss Map, Examples and Application

TL;DR

This paper explores the properties of amoebas and their contours in relation to the logarithmic Gauss map, connecting algebraic singularity theory and computational methods to analyze critical points and configurations.

Contribution

It unifies various perspectives on amoebas, singularities, and the logarithmic Gauss map, and demonstrates computational approaches to identifying critical points.

Findings

Computed examples using SINGULAR highlighting computational challenges.

Identified the importance of real or rational solutions for critical point analysis.

Connected amoeba configurations with algebraic singularity classifications.

Abstract

The study of hypersurfaces in a torus leads to the beautiful zoo of amoebas and their contours, whose possible configurations are seen from combinatorial data. There is a deep connection to the logarithmic Gauss map and its critical points. The theory has a lot of applications in many directions. In this report we recall basic notions and results from the theory of amoebas, show some connection to algebraic singularity theory and discuss some consequences from the well known classification of singularities to this subject. Moreover, we have tried to compute some examples using the computer algebra system SINGULAR and discuss different possibilities and their effectivity to compute the critical points. Here we meet an essential obstacle: Relevant examples need real or even rational solutions, which are found only by chance. We have tried to unify different views to that subject.