Vol. 317, No. 1, 2022

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Describing amoebas

Mounir Nisse and Frank Sottile

Vol. 317 (2022), No. 1, 187–205
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 results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semialgebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba which is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semialgebraic sets.

Keywords
amoeba, coamoeba, tropical geometry
Mathematical Subject Classification
Primary: 14T99, 14T05, 32A60
Milestones
Received: 16 February 2021
Revised: 25 June 2021
Accepted: 2 July 2021
Published: 19 June 2022
Authors
Mounir Nisse
Department of Mathematics
Xiamen University Malaysia
Sepang
Malaysia
Frank Sottile
Department of Mathematics
Texas A&M University
College Station, TX
United States