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).