How does the tropicalization of a variety change as we change the valuation of a field? Let $K$ be an algebraically closed field and let $V$ be a subvariety of the algebraic torus.
The tropicalization of $V$ actually depends on what valuation we're using on the field $K$. How does the tropicalization of $V$ change as we change the valuation function?
 A: Roughly speaking, the unbounded parts of the two tropicalizations will look the same, and you have almost no control over the bounded interior. More precisely, for $K \subset \mathbb{R}^n$, define $K_{\infty} = \lim_{t \to 0^{+}} t \cdot K$, where the limit can be taken in the sense of Hausdorff distance between subsets of $\mathbb{R}^n$. Knowing $(\mathrm{Trop} \ X)_{\infty}$ is basically equivalent to knowing the Chow polytope of $X$, which controls the cohomology class of the closure of $X$ in every toric embedding. When $X$ is a hypersurface $\{ F=0 \}$, the Chow polytope can be described concretely as the Newton polytope of $F$. The hypersurface case is easy enough to work out by hand; Alex Fink's thesis studies the general case.
I don't have a precise statement of claim that you have little control over the interior, but let's think about the hypersurface case. Let $v_1$ and $v_2$ be two valuations on $K$. Let $F_1$ and $F_2$ be two Laurent polynomials, with coefficients in $K_{v_1}$ and $K_{v_2}$, and with the same Newton polytope $\Delta$. Then we can find some Laurent polynomial $G$ with Newton polytope $\Delta$ and coefficients which are $v_i$-adically close to $F_i$. Then $\mathrm{Trop} \ G$ for the valuation $v_i$ will look like $\mathrm{Trop} \ F_i$, and we don't get any relations between the two beyond what we can deduce from having the same Newton polytope.
That are some interesting things you can say when $X$ is defined over a global field $K$ and you consider the set of all tropicalizations for all valuations of $K$. See Sam Payne's paper on adelic amoebas and the references cited therein.
