Amoebas of maximal area.

By G. Mikhalkin, H. Rullg̊ard. 2000

93 citations recorded by the source corpus.

To any algebraic curve A in (*)2 one may associate a closed infinite region A in 2 called the amoeba of A. The amoebas of different curves of the same degree come in different shapes and sizes. All ...

Subfields: Computational Mechanics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/897630bf2c4296994c18fecf5087d1b0bff2eec8

How researchers see this paper

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

In Computational Mechanics

1. Richard Kenyon
2. Mikael Passare
3. A. Okounkov
4. Eugenii Shustin
5. Scott Sheffield
6. H. Rullg̊ard
7. Zur Izhakian
8. Louis H. Rowen

In Computational Theory and Mathematics

1. Zur Izhakian
2. Louis H. Rowen
3. Eugenii Shustin
4. A. Okounkov
5. Richard Kenyon
6. G. Mikhalkin
7. Mikael Passare
8. H. Rullg̊ard

In Geometry and Topology

1. Eugenii Shustin
2. G. Mikhalkin
3. Zur Izhakian
4. Louis H. Rowen
5. Richard Kenyon
6. A. Okounkov
7. Mikael Passare
8. H. Rullg̊ard

Citations in the corpus

• Dimers and cluster integrable systems (2011, background): “nition via 2−1 covering of its amoeba. Recall that the amoeba is the image of the curve under the map C ∗×C →R2, (z 1,z2) →(log|z1|,log|z2|). Another characterization of simple Harnack curves, due to [MR00], is that they are the curves which maximize the area of their amoeba among all curves with gi”
• Amoebas, Monge-Ampère measures, and triangulations of the Newton polytope. (2004, methodology): “This estimate was used in [8] to give a new characterization of so-called Harnack curves, which are fundamental for Hilbert's sixteenth problem in real algebraic geometry.”
• 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”
• 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].”
• The Tropical Rank of a Tropical Matrix (2006, background): “is the real numbers equipped by the operations of maximum and summation – addition and multiplication [12, 25], respectively. This semi-ring is usually extended by −∞, the neutral element of addition [18, 19]. The basics of the corresponding tropical algebraic-geometric formalism have been presented”
• Dimers and amoebae (2003, methodology): “The precise topological definition of a Harnack curve can be found in [14]; here we will use the following alternative characterization of a Harnack curve obtained in [ 15 ].”
• Amoebas of Algebraic Varieties and Tropical Geometry (2004, background): “k, Itenberg-Viro [12], [16]). For any ∆ there exists a curve RV ⊂ (R∗)2 which is torically maximal and has ∆ as its Newton polygon. As in Definition 3.11 we call such curves simple Harnack curves, cf. [27]. 3.5.3. Geometric properties of algebraic curves in (R∗)2. It turns out that the simple Harnack”
• Tropical Geometry and its applications (2006, unknown): “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 intermediate link be”