Amoebas of algebraic varieties

TL;DR

This survey reviews the development and applications of amoebas of algebraic varieties, highlighting their role in understanding the topology and geometry of algebraic structures across multiple mathematical disciplines.

Contribution

It summarizes current knowledge on amoebas, their properties, and their applications in algebraic geometry and related fields, providing a comprehensive overview.

Findings

Amoebas serve as powerful tools in studying the topology of algebraic varieties.

They have applications in real algebraic geometry and combinatorics.

The survey consolidates recent advances and outlines future directions.

Abstract

The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some traces of amoebas were appearing from time to time, even before the formal introduction, as auxiliary tools in several problems. After 1994 amoebas have been seen and studied in several areas of mathematics, from algebraic geometry and topology to complex analysis and combinatorics. In particular, amoebas provided a very powerful tool for studying topology of algebraic varieties. This survey aims to summarize the current state of knowledge about amoebas and to outline the applications to real algebraic geometry and adjacent areas. Most proofs are omitted here. An expanded version of this survey is currently under preparation jointly with Oleg Viro.