Amoebas of algebraic varieties

Grigory Mikhalkin

Published 2001-08-31Version 1

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.