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Recently I heard of a recent field in mathematics called tropical geometry. Having read the wiki page on it it seems like it is combinatorial algebraic geometry.

My question is what are the benefits of applying tropical geometry to problems in algebraic geometry? Are there examples of theorems in algebraic geometry where a proof was made much simpler using tropical geometry? Or are there any conjectures in algebraic geometry that was proved using tropical geometry?

Also does anyone know the motivation behind why it was developed?

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Have you looked at the link arxiv.org/abs/math.AG/0601041 from the wikipedia page, in particular at section 7? Since Mikhalkin himself is not on this site, you will not get a better answer than his survey article(s). –  Alex B. Jun 8 '12 at 5:36
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Since you're interested in number theory, you might look at some papers of Joe Rabinoff's. –  Dylan Moreland Jun 8 '12 at 5:47
    
I know the proof of how many general degree $d$ curves there are passing through $3d-1$ points in general position is made much simpler by using tropical geometry - you can show the answer will be the same for tropical degree $d$ curves, and they're easier to count. The lack of an answer to the tropical inverse problem (which tropical varieties are tropicalizations of algebraic varieties) makes it difficult to go from tropical to algebraic, but I believe there are examples of algebraic theorems proved like this, and Mikhalkin is probably a good place to look. –  Matt Pressland Jun 8 '12 at 8:45
    
@AlexB. Thanks for the link. It was certainly helpful. –  Eugene Jun 8 '12 at 20:18
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up vote 2 down vote accepted

Here is a link to a Mathoverflow thread that gives several references for someone wanting to learn tropical algebraic geometry, including to the recent book of Maclagan and Sturmfels. Much of the motivation for tropical algebraic geometry can be found by perusing the papers mentioned there.

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I was just about to mention Diane Maclagan too! I knew she was interested in this area, but I didn't know about the book. –  Tara B Jun 8 '12 at 20:49
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