Amoebas of Algebraic Varieties and Tropical Geometry

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content of Part 2. We pay special attention to tropical curves. Both parts also include relevant applications of the theories. Part 1 of this survey is a revised and updated version of an earlier prepreint of 2001.

[1]  A. Harnack,et al.  Ueber die Vieltheiligkeit der ebenen algebraischen Curven , 1876 .

[2]  Nota di Luigi Brusotti.  Curve generatrici e curve aggregate nella costruzione di curve piane d’ordine assegnato dotate del massimo numero di circuiti , 1916 .

[3]  I. Petrowsky,et al.  On the Topology of Real Plane Algebraic Curves , 1938 .

[4]  V. Arnold.  On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms , 1971 .

[5]  George M. Bergman,et al.  The logarithmic limit-set of an algebraic variety , 1971 .

[6]  V. Rokhlin,et al.  Congruences modulo 16 in Hilbert's sixtieth problem , 1972 .

[7]  V. Rokhlin.  Congruence modulo 16 in Hilbert's sixteenth problem. II , 1973 .

[8]  D. N. Bernshtein.  The number of roots of a system of equations , 1975 .

[9]  Joe Harris,et al.  Real algebraic curves , 1981 .

[10]  M. Atiyah,et al.  Angular momentum, convex Polyhedra and Algebraic Geometry , 1983, Proceedings of the Edinburgh Mathematical Society.

[11]  V. Kharlamov,et al.  Classification of nonsingular surfaces of degree 4 in RP3 with respect to rigid isotopies , 1984 .

[12]  V. P. Maslov,et al.  On a new superposition principle for optimization problem , 1986 .

[13]  Ian Stewart,et al.  Hilbert's sixteenth problem , 1987, Nature.

[14]  M. Kapranov,et al.  A characterization ofA-discriminantal hypersurfaces in terms of the logarithmic Gauss map , 1991 .

[15]  Eugenii Shustin,et al.  Topology of real plane algebraic curves , 1992 .

[16]  I. M. Gelʹfand,et al.  Discriminants, Resultants, and Multidimensional Determinants , 1994 .

[17]  M. Kontsevich,et al.  Gromov-Witten classes, quantum cohomology, and enumerative geometry , 1994 .

[18]  O. Viro,et al.  Patchworking algebraic curves disproves the ragsdale conjecture , 1996 .

[19]  Branes, superpotentials and superconformal fixed points , 1997, hep-th/9704170.

[20]  Webs of (p,q) 5-branes, five dimensional field theories and grid diagrams , 1997, hep-th/9710116.

[21]  H. Hofer,et al.  Introduction to Symplectic Field Theory , 2000, math/0010059.

[22]  Amoebas of maximal area. , 2000, math/0010087.

[23]  Hans Rullgård.  Stratification des espaces de polynômes de Laurent et la structure de leurs amibes , 2000 .

[24]  Mikael Passare,et al.  Laurent determinants and arrangements of hyperplane amoebas , 2000 .

[25]  G. Mikhalkin.  Real algebraic curves, the moment map and amoebas , 2000 .

[26]  Yan Soibelman,et al.  Homological mirror symmetry and torus fibrations , 2000 .

[27]  Oleg Viro,et al.  Dequantization of Real Algebraic Geometry on Logarithmic Paper , 2000, math/0005163.

[28]  F. Bourgeois.  A Morse-Bott approach to contact homology , 2002 .

[29]  Grigory Mikhalkin.  Counting curves via lattice paths in polygons , 2002 .

[30]  Grigory Mikhalkin.  Decomposition into pairs-of-pants for complex algebraic hypersurfaces , 2002 .

[31]  E. Shustin,et al.  Welschinger invariant and enumeration of real plane rational curves , 2003, math/0303378.

[32]  R. Kenyon,et al.  Dimers and amoebae , 2003, math-ph/0311005.

[33]  Andrei Okounkov,et al.  Planar dimers and Harnack curves , 2003 .

[34]  B. Sturmfels,et al.  First steps in tropical geometry , 2003, math/0306366.

[35]  Mikael Passare,et al.  Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope. , 2004 .

[36]  J. Pin.  Tropical Semirings Jean-Eric Pin , 2005 .

[37]  H. Rullg̊ard.  Polynomial amoebas and convexity , 2022 .