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Could someone tell me if it is possible to do tropical geometry with NO knowledge(or with very few) of algebraic geometry (a la Hartshorne)? By "do tropical geometry" I mean, to understand the important theorems as well as read literature.

In other words, suppose that I have a background in combinatorics, graphs etc., and am looking for a problem (and its solution) in Tropical Geometry but perhaps I do not really know much about schemes, sheaves, Cech Cohomology?. Would I have a change to succeed? It makes sense since Hartshorne's AG may take some time to master.

Concrete arguments against/in favour of would be highly appreciated.

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3 Answers 3

Yes, it is possible. People who have done excellent tropical work with little Algebraic Geometry background include Federico Ardila, Michael Joswig and Josephine Yu. (I hope I won't insult any of these people by saying that they do not strike me as having much algebraic geometry.)

However, I have had bad luck introducing people to tropical geometry without talking about valued fields, Grobner degenerations, toric varieties and the other algebraic technology. I can give a nice colloquium talk or write a nice expository paper where I gloss over this material. But this leaves the reader without an intuition to figure out which questions are reasonable to ask, or any idea of where nontrivial results might come from. This is especially true because so much tropical work right now is not solving specific problems formulated by experts, but in finding the definitions and theorems to make precise the phenomena which people have observed.

ADDED I also like Ben's answer. There are parts of tropical geometry which use very serious algebraic geometry, but there are also parts where it is being used for motivation and intuition. You could probably get a lot of what you need from Cox-Little-O'Shea, Fulton's "Algebraic Curves" and some good reference on Grassmannians and hyperplane arrangements.

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I have a not-so-concrete argument against, which is that graduate school is not the time to try to learn things "on the cheap." This is more understandable later in academic life, but graduate school is supposed to be a time in which you broaden your horizons.

On the other hand, I'm not sure if Hartshorne is the place to start; getting a decent understanding of affine and projective varieties over C is would be a bit simpler (maybe starting with any part of Cox, Little and O'Shea you don't understand, followed by Miller and Sturmfels?).

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I had in mind something like David's answer. Meaning, great tropical geometry (done by great mathematicians) where heavy machinery of stacks or so is not the problem solver. –  Csar Lozano Huerta Dec 3 '09 at 6:54

Recently, M. Aschenbrenner, D. Lippel, and S. Starchenko used a nonstandard analysis approach to reprove a basic theorem in the subject (the Bieri-Groves Theorem relating tropical varieties to tropical amoebas). Here is a nice outline of the argument, which is elementary and completely free of contemporary-style algebraic geometry.

Caveat: I know very little about tropical geometry and I have no idea whether these model-theoretic methods can be pushed further to prove other results in the field.

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