The general result in tropical geometry is $K$ algebraically closed valued field $I$ ideal of $K[x_1, \cdots, x_n]$ $V(I) = \lbrace \bar{a}\in K^n: f(\bar{a})=0 \text{ for all } f \in I \rbrace$.
Let $f \in K[x_1,\cdots,x_n]$ such that $f(X) = \sum a_j X^j$ $j \in \mathbb{N}^n$ then $\operatorname{Trop}(f): \mathbb{R}^n \rightarrow \mathbb{R}$ $X\mapsto \max\lbrace-v(a_j)+\langle X,j\rangle\rbrace$ where $v$ is the valuation map.
A root of $\operatorname{Trop}(f)$ is a tuple $w$ such that the maximum of $\lbrace -v(a_j) +\langle w, j \rangle \rbrace$ is attained at least twice.
The fundamental theorem of tropical geometry states $v(V(I)) = \operatorname{Roots}(\operatorname{Trop}(I))$ (actually $\text{the topological closure of } v(V(I)) = \operatorname{Roots}(\operatorname{Trop}(I))$, or $v(V(I)) = \operatorname{Roots}(\operatorname{Trop}(I))\cap \mathbb{Q}^n$ in the case of the Puiseux field). Trivially $v(V(I)) \subseteq \operatorname{Roots}( \operatorname{Trop}(I))$, nevertheless the converse is not so easy to prove.
My question goes on how to prove the converse for $K$ the field of Puiseux series. Maybe an approach through Hensel's Lemma.