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?
Best Answer
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.