Recall that an open subset $U$ of $X$ is called very affine if $U$ has a closed embedding into a multiplicative torus. Clearly, the following conditions are equivalent for an open affine subset $U$ of $X$:

• (a)

  $U$ is very affine;

• (b)

  ${\mathscr{O}}(U)$ is generated as a $K$-algebra by ${\mathscr{O}}(U)^{\times}$;

• (c)

  the canonical moment map $\varphi_{U}$ from 4.12 is a closed embedding.

The intersection of two very affine open subsets is again very affine (see the proof of Proposition 4.16). Moreover, the very affine open subsets of $X$ form a basis for the Zariski topology. We conclude that all local considerations can be done using very affine open subsets.

On a very affine open subset, we will almost always use the canonical moment map $\varphi_{U}:U\rightarrow T_{U}$ which is a closed embedding by the above. To simplify the notation, we will set ${\rm Trop}(U)$ for the tropical variety of $U$ in $T_{U}$. It is a tropical cycle in $(N_{U})_{\mathbb{R}}$, where $N_{U}$ is the dual abelian group of $M_{U}$. The tropicalization map will be denoted by ${\rm trop}_{U}:=(\varphi_{U})_{\rm trop}:{U^{\rm an}}\rightarrow(N_{U})_{\mathbb{R}}$. Recall that $\varphi_{U}$ is only determined up to translation by an element of $T_{U}(K)$ and hence ${\rm trop}_{U}$ and ${\rm Trop}(U)$ are only canonical up to an affine translation. This ambiguity is no problem as our constructions will be compatible with affine translations.