Tropical bases

Let $I \subset \mathbb{C}[x_1, \ldots, x_n]$ be an ideal. The problem is to characterize/compute subsets $J$ of $I$ which suffice to define the tropical variety $\mathcal{T}(I)$, i.e. $\mathcal{T}(I) = \cap_{j \in J} \mathcal{T}(j)$.

Theorem. The $3 \times 3$-minors of an $n \times n$-matrix of indeterminates (which are not a not a universal Gröbner basis) suffice to define the tropical variety of that ideal.

Question (Sturmfels): Do the $4 \times 4$-minors of a $5 \times 5$-matrix of indeterminates (which are far from a universal Gröbner basis) suffice to define the tropical variety?

Question: Find a characterization of a (smaller) sufficient set (which should be easier/better to compute).

(contributed by Rekha Thomas)

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