Tropical Algebraic Geometry Keyvan Yaghmayi University of Utah Advisor: Prof. Tommaso De Fernex

Field of interest and the problem There are different approaches to tropical algebraic geometry. In the following I describe briefly three approaches: the geometry over max-plus semifield, the Maslov dequantization of classical algebraic geometry, and the valuation approach. We are interested to study the valuation approach and also understand the relation with Berkovich spaces.

Tropical algebraic geometry Example 1 (integer linear programming): Let be a matrix of non-negative

The most straightforward way to tropical algebraic geometry is doing algebraic geometry on integers, let be a row vector with real entries, and let be a column tropical semifield where the tropical sum is maximum and the tropical vector with non-negative integer entries. Our task is to find a non-negative integer column vector product is the usual addition: . which solves the following optimization problem:

Ideally, one may hope that every construction in algebraic geometry should have a tropical Minimize subject to and . (1) counterpart and thus, obtain results in algebraic geometry by looking at the tropical picture and Let us further assume that all columns of the matrix sum to the same number and that then trying to transfer the results back to the original setting. . This assumption is convenient because it ensures that all feasible solutions of (1) satisfy . We can solve the integer programming problem (1) using tropical arithmetic We can look at as the Maslov dequantization of the semifield . Let for . as follows. Let be indeterminates and consider the expression The bijection given by induces a semifield structure on such that

is an isomorphism. So, is a semifield and for we have: (2)

The optimal value of (1) is the coefficient of the monomial in the th Proposition 1. Thus, . we can power of the tropical polynomial (2). combine with norm and consider the tropical algebraic geometry as the Example 2 (Gromov-Witten invariants): the genus of a curve of degree in is

Maslov dequantization of classical algebraic geometry. and the set of all curves of degree form a projective space of dimension Now it is easy to see the relation with amoebas of an algebraic variety. The amoeba of a variety

. The number of irreducible curves of degree an genus is so . We associate a family of amoebas to and then we define the tropical variety, , associated to that pass through general point in are called Gromov-Witten numbers and are denoted the variety to be the limit of this family . by . Mikhalkin in [M05] showed that the Gromov-Witten numbers can be found tropically:

Remark: In some texts like [IMS07] the associated family of amoebas is the image under Propesition 2. The number of simple tropical curves of degree and genus that pass through the map defined by . By change of – general points in , where each curve is counted with its multiplicity, is equal to the variable we see if and only if . Gromov-Witten number of the complex projective plane .

For valuation approach let be the field of Laurent series and = References [B90] Vladimir G. Berkovich, Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, be the field of Puiseux series over . For we define the Number 33 of Mathematical Surveys and Monographs, American Mathematical Society, 1990. valuation on and also consider given by [IMS07] Ilia Itenberg, Grigory Mikhalkin, and Eugenii Shustin. Tropical Algebraic Geometry, Volume 35 of Oberwolfach Seminars. Birkhauser Verlag AG, 2007. . Let be an ideal in we associate to a [MS09] Diane Maclagan, Bernd Sturmfels. Introduction to Tropical Geometry. Book in Preparation, tropical variety where is the very affine variety corresponding November 4, 2009. to and the bar is topological closure in . It is easy to see . [V01] Oleg. Ya. Viro. Dequantization of Real Algebraic Geometry on a Logarithmic Paper. Proceedings Application of the 3rd European Congress of Mathematicians, Birkhauser, Progress in Math, 201, (2001), 135-146. Here I give two examples: the first one shows how tropical geometry was developed in the context [M05] Grigory Mikhalkin, Enumerative Tropical Algebraic Geometry in . American Mathematical of discrete mathematics and optimization. The second shows its power and usefulness in algebraic Society, 18, (2005), 313-377. geometry.