# What is Tropicalization, and how is it applied

My question is:

## What is Tropicalization, how is it done, and what are some basic applications of it?

### motivation

I am interested especially in how questions about enumerative algebraic geometry translate into the tropical world and how information obtained in the tropicalization reflects back on the original question.

This rather general question is similar to earlier questions I asked about integrable systems and about categorification. There are some relevant Wikipedia articles but while interesting I don't find them sufficient, and several point of views from MO participants can be enlightening and lead to a good resource.

A related MO question Why tropical geometry?

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Isn't this basically a request for a whole book? Such as this one: homepages.warwick.ac.uk/staff/D.Maclagan/papers/… ? – Alex B. Oct 10 '13 at 15:51
Alex B. no i don't think so. Look at the other two problems linked in the question. – Gil Kalai Oct 11 '13 at 6:46
Maxim Kontsevich has just given three Coxeter lectures at the Fields Institute (videos online). Lecture 1: Elementary intro to tropical mathematics" is a convincing argument for the basic fundamental nature of tropical mathematics. Lectures 2,3 contained many references to tropicalization which appear part of some much larger project and vision. – J. Martel Oct 19 '13 at 15:20
Here are are the links to Kontsevich's lectures, mentioned by J. Martel fields.utoronto.ca/video-archive/event/22/2013 – j.c. Oct 19 '13 at 16:56

The answer is easily found in the literature. Here is a list of some general expositions:

MR2902202 Itenberg, I.; Mikhalkin, G. Geometry in the tropical limit. Math. Semesterber. 59 (2012), no. 1, 57–73.

MR2305295 Mikhalkin, Grigory What is…a tropical curve? Notices Amer. Math. Soc. 54 (2007), no. 4, 511–513.

MR2275625 Mikhalkin, Grigory Tropical geometry and its applications. International Congress of Mathematicians. Vol. II, 827–852, Eur. Math. Soc., Zürich, 2006.

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Thanks, Alexandre! – Gil Kalai Oct 20 '13 at 22:26

One may also consult the book-in-progress by Diane Maclagan and Bernd Sturmfels at this link: Introduction to Tropical Geometry. The first five chapters are available as of this summer.

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(I didn't notice that Alex B. earlier cited the same link.) – Joseph O'Rourke Oct 21 '13 at 0:20
Thank you, Joe ! – Gil Kalai Nov 3 '13 at 15:29