Recognition problems

1. How to recognize whether a a pure $d$-complex is a tropical variety?
2. Characterize the image of linear subspaces w.r.t. under the phase map (``phlat''). i.e., characterize $\pi(L)$ where $L \subset \mathbb{C}^*$ is a linear subspace, and $\pi$ is the phase map $\pi : \mathbb{C}^* \to T^n$ ($T^n$: $n$-dimensional torus).

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