Amoebas of Algebraic Varieties and Tropical Geometry

By G. Mikhalkin. 2004

190 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ee220cb31bb299a8851e4537c2db07afcdf9e036

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Citations in the corpus

Affine Structures and Non-Archimedean Analytic Spaces (2004, background): “ideas is a subject of the so-called “tropical geometry” (see e.g. [ Mi ]).”
Discrete conformal maps and ideal hyperbolic polyhedra (2010, background): “(See in particular Kenyon’s survey article on dimers [14] and Mikhalkin’s survey article on amoebas [21].”
Brane tilings and their applications (2008, background): “The resulting geometry is called “tropical geometry”(26), which is an active area of research in mathematics (see [87, 88] for reviews).”
A tropical approach to secant dimensions (2006, methodology): “o secant dimensions, based on tropical geometry. Tropical geometry is well known as a tool for transforming algebraicgeometric questions into polyhedral-combinatorial ones. Recommended references are [7, 16, 22, 24, 25, 26] and the references therein—however, all background in tropical geometry need”
Tropical varieties for non-archimedean analytic spaces (2006, background): “algebraic geometry. We refer to [ Mi ] for a survey of this relatively new area of research.”
Dimer models from mirror symmetry and quivering amoebae (2005, background): “h to the toric diagram, i.e., the (p,q)-web. For reference we include the toric diagram and the Riemann surface Σ in (c). Let us discuss the projection (6.2) in detail ﬁrst. The reader is referred to [21, 39, 40, 41, 25]. The formal deﬁnition applies to any variety. Let V ⊂ (C∗)n be an algebraic var”
Amoebas, nonnegative polynomials and sums of squares supported on circuits (2014, background): “having their nature in complex algebraic geometry with applications in various mathematical subjects including complex analysis [10,11], the topology of real algebraic curves [24], dynamical systems [8], dimers/crystal shapes [18], and in particular with strong connections to tropical geometry, see ”
On the tropical Torelli map (2009, background): “For an introduction to tropical geometry, see the surveys [39,55,52,22,41,28,43], or the books in preparation [44,34].”