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. The formula was announced earlier in [17]. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows to replace complex toric varieties with the real space Rn and holomorphic curves with certain piecewise-linear graphs there.

/pdf/enumerative-tropical-algebraic-geometry-in-3vgytnxpm7.pdf

Enumerative tropical algebraic geometry in ℝ

Tropical islands Building blocks Tropical varieties Tropical rain forest Tropical garden Toric connections Bibliography Index

Introduction to Tropical Geometry

Enumerative tropical algebraic geometry

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 talk outlines some basic notions of this area and surveys some of its applications
for the problems in classical (real and complex) geometry.

http://www.icm2006.org/proceedings/Vol_II/contents/ICM_Vol_2_40.pdf

Tropical geometry and its applications

In this paper we study surfaces in R
 3 that arise as limit shapes in random surface models related to planar dimers. These limit shapes are surface tension minimizers, that is, they minimize a functional of the form ∫σ(∇h) dx
 dy among all Lipschitz functions h taking given values on the boundary of the domain. The surface tension σ has singularities and is not strictly convex, which leads to formation of facets and edges in the limit shapes. We find a change of variables that reduces the Euler–Lagrange equation for the variational problem to the complex inviscid Burgers equation (complex Hopf equation). The equation can thus be solved in terms of an arbitrary holomorphic function, which is somewhat similar in spirit to Weierstrass parametrization of minimal surfaces. We further show that for a natural dense set of boundary conditions, the holomorphic function in question is, in fact, algebraic. The tools of algebraic geometry can thus be brought in to study the minimizers and, especially, the formation of their singularities. This is illustrated by several explicitly computed examples.

/pdf/limit-shapes-and-the-complex-burgers-equation-19thov7ch0.pdf

Limit shapes and the complex Burgers equation

The amoeba of a holomorphic function $f$ is, by definition, the image in $\mathbf{R}^n$ of the zero locus of $f$ under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function $N_f$ with the property that its Monge-Ampere mass is concentrated to the amoeba of $f$ We obtain results of two kinds; by approximating $N_f$ with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of $f$; by computing the Monge-Ampere measure, we find sharp bounds for the area of amoebas in $\mathbf{R}^n$. We also consider systems of functions $f_{1},\dots,f_{n}$ and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.

Mikael Passare

Papers

Amoebas, Monge-Ampàre measures, and triangulations of the Newton polytope