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What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series $\mathbb{C}((t))$ what can be said about $\Gamma ^n$ where $\Gamma$ is the valuation group.

That is what pairs of properties $(X,X')$ such that $X$ is valid on an algebraic variety then $X'$ is valid on the tropicalization, and $X$ is a first order property in the language $L={ 0,1,+,*,U,|}$ where $U$ stands for the valuation ring, and $x|y \leftrightarrow \exists z U(z) _\wedge xz=y$

A silly example would be the following: One can prove that when the support (the set of exponents corresponding to non zero coefficients) of a polynomial $f$ in two variables is equal to {$(i,j)\in \mathbb{N}^2 | i+j \leq d$} for some $d\in \mathbb{Z}^+$ and the support of another polynomial $g$ is {$(i,j)\in \mathbb{N}^2 | i+j \leq c$} for another $c\in \mathbb{Z}^+$ then the tropical curves generated by those polynomials intersects in exactly $cd$ points or in infinitely many(Bézout).

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What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series $\mathbb{C}((t))$ what can be said about $\Gamma ^n$ where $\Gamma$ is the valuation group.

That is what pairs of properties $(X,X')$ such that $X$ is valid on an algebraic variety then $X'$ is valid on the tropicalization, and $X$ is a first order property in the language $L={ 0,1,+,*,U,|}$ where $U$ stands for the valuation ring, and $x|y \leftrightarrow \exists z U(z) _\wedge xz=y$

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Tropical Properties From Algebraic Geometry

What properties of tropical geometry can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series $\mathbb{C}((t))$ what can be said about $\Gamma ^n$ where $\Gamma$ is the valuation group.