Describing Amoebas

TL;DR

This paper explores the geometric and algebraic properties of amoebas, images of algebraic varieties under the logarithmic map, focusing on their dimensions, semi-algebraic descriptions, and intersections.

Contribution

It establishes new results on amoeba dimensions, semi-algebraic representations, and characterizations of amoebas of lines, and poses open problems for further research.

Findings

Non-full-dimensional amoebas are not finite intersections of hypersurface amoebas if the variety is nondegenerate.

Explicit semi-algebraic descriptions of amoebas of lines are provided.

Open problems include determining amoeba dimensions and characterizing amoebas as semi-algebraic sets.

Abstract

An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semi-algebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semi-algebraic sets.