Real algebraic curves, the moment map and amoebas

By G. Mikhalkin. 2000

206 citations recorded by the source corpus.

In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexiflcations.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f66cf95a355bc67db2886ec96deace09f98f172d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Dimer models from mirror symmetry and quivering amoebae (2005, background): “diagram of F0 is indicated. Now, this curve P(z,w)−W = 0 is a genus g (punctured) Riemann surface, which we will denote as Σ W, ﬁbred over each point W. The genus is prescribed by the simple relation [21, 22] g = number of internal points in the toric diagram. (2.5) Of particular importance to us is”
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 ”
Random polynomials with prescribed Newton polytope (2002, background): “(The correspondence is given by [ Mi1 , §3.1] or [FPT].) Hence the number of tentacles of A equals the number of points of ∂P ∩ Z; this number is called the length of ∂P. (For example, the 12 tentacles in Figure 5 correspond to the 12 lattice points of ∂(4�).) Each tentacle corresponds to a segment ”
Brief introduction to tropical geometry (2015, methodology): “oduction of amoebas due to I.M. Gelfand, M.M. Kapranov and A.V. Zelevinsky [GKZ94], further relations between amoebas and real algebraic curves were found by O. Viro and G. Mikhalkin (see [Vir01] and [Mik00]). We discuss here only a particular case of the general patchworking theorem. This case, cal”
Enumerative tropical algebraic geometry in R^2 (2003, background): “Another particularly interesting case is if F = C with the standard norm ||z|| = √ zz̄ (see [6],[16],[19], etc.”
Amoebas of maximal area. (2000, unknown): “Note that by Corollary 2, Area(A) = π2 Area(∆) if and only if µ = (λ/π2)|A.”
Moduli of tropical plane curves (2014, background): “Nondegenerate polynomials are useful for studying many subjects in algebraic geometry, including singularity theory [23], the theory of sparse resultants [17], and topology of real algebraic curves [27].”
Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry (2002, background): “Geometry of such amoebas and their applications have been studied in [5, 6, 7, 12, 13, 14, 17, 18].”
Tropical Geometry and its applications (2006, unknown): “onsider the map Logt : (C ×)n → Rn corresponding to taking the logarithm with the base t > 1. Amoebas themselves have proved to be a very useful tool in several areas of Mathematics, see e.g. [4], [15], [16], [20], [24], [25]. However, for the purposes of this talk we only use them as an intermed”
Planar dimers and Harnack curves (2003, background): “A G ] 1 8 O ct 2 00 4 Planar dimers and Harnack curves Richard Kenyon and Andrei Okounkov October 2003”
Amoebas of Algebraic Varieties and Tropical Geometry (2004, background): “Figure 2. [ 26 ] An amoeba A and a compactified amoeba ¯”
Enumerative tropical algebraic geometry (2003, background): “The polynomial f∆ ′ min is well-defined up to a multiplication by a constant.”
Analytic Combinatorics in Several Variables (2013, unknown): “Mikhalkin (2000); Grigory Mikhalkin (2004).”
Dimers and cluster integrable systems (2011, background): “rizes connections on a minimal bipartite graph Γ associated with Nwhose face weights are positive numbers, and does not depend on the choice of Γ. Simple Harnack curves were deﬁned by G. Mikhalkin in [Mikh00] as real curves P(z1,z2) = 0 whose real components satisfy certain topological constraints. ”