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Enumerative tropical algebraic geometry

- G. Mikhalkin
- Mathematics
- 31 December 2003

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon.… Expand

Enumerative tropical algebraic geometry in R^2

- G. Mikhalkin
- Mathematics, Physics
- 31 December 2003

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon.… Expand

Tropical geometry and its applications

- G. Mikhalkin
- Mathematics
- 3 January 2006

From a formal perspective tropical geometry can be viewed as a branch of geometry
manipulating with certain piecewise-linear objects that take over the role of classical algebraic
varieties. This… Expand

Tropical curves, their Jacobians and Theta functions

- G. Mikhalkin, Ilia Zharkov
- Mathematics
- 11 December 2006

We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and… Expand

ENUMERATIVE TROPICAL ALGEBRAIC GEOMETRY IN R

- G. Mikhalkin
- 2004

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon.… Expand

Real algebraic curves, the moment map and amoebas

- G. Mikhalkin
- Mathematics
- 2 October 2000

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… Expand

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

- G. Mikhalkin
- Mathematics
- 1 May 2002

Abstract It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth… Expand

Labeled floor diagrams for plane curves

- S. Fomin, G. Mikhalkin
- Mathematics
- 20 June 2009

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative)… Expand

Enumeration of curves via floor diagrams

- Erwan Brugall'e, G. Mikhalkin
- Mathematics
- 1 June 2007

In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.

Amoebas of Algebraic Varieties and Tropical Geometry

- G. Mikhalkin
- Mathematics
- 29 February 2004

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… Expand

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