Idempotent geometry

Questions:

1.

Is it possible to construct a version of algebraic geometry over a class of algebraically closed idempotent semifields (not only tropical semifields)?

Remark: A simple criterion for an idempotent semifield to be algebraically closed is proved in the paper of G. Shpiz ``Solving algebraic equations in idempotent semifields'', Uspekhi Mat. Nauk, v.55, #5 (2000), p.185-186 (in Russian; there is an English translation in Russian Mathematical Surveys, 2000). There are many examples of algebraically closed idempotent semifields. For example, some standard linear function spaces and all the Banach lattices generate algebraically closed idempotent semifields; see, e.g., the paper of G.L. Litvinov, V.P. Maslov, and G.B. Shpiz ``Idempotent functional analysis: an algebraic approach'', Math. Notes, v.69, #5 (2001), p.758-797.

2.

Is it possible to define a notion of an abstract algebraic (not only affine or projective) variety over tropical and idempotent semifields?

3.

Is it possible to define idempotent/tropical versions of such concepts as regular functions and regular maps to get a natural category of idempotent/tropical ``affine'' algebraic varieties? Is it possible to construct a natural correspondence between this category and a category of idempotent semirings of functions in the spirit of the traditional algebraic geometry?

4.

It would be useful to define tropical/idempotent versions of such notions as algebraic equations and ideals of affine algebraic varieties in such a way that points and subvarieties correspond to analogs of ideals.

5.

It would be useful to describe tropical/idempotent versions of such notions as prime ideals and irreducible varieties. How to investigate the corresponding decomposition into irreducible components?

6.

It would be nice to construct dequantization procedures for a natural correspondence between traditional algebraic varieties and tropical varieties? Is it possible to construct something like a functor (``almost functor'') between the corresponding categories?

(contributed by G.L. Litvinov, in cooperation with G.B. Shpiz)

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