Tropically smooth after change of coordinates

# Setup

Let $K$ be a valued field. A tropical cubic surface $\mathcal T(f)$ defined by a polynomial $f \in K[x,y,z]$ is smooth if the regular subdivision induced by the valuation of the coefficients on the Newton polytope of $f$ is a unimodular triangulation, i.e., the maximal cells in the subdivision are simplices of minimal volume $\frac{1}{6}$, see Chapter 4.5 of [1].

# Question

How can we decide if a given polynomial defines a smooth tropical surface after a linear transformation of $\mathbb P^3_K$? How can we find the corresponding 4×4 matrix?

# Endgame:

• Introduce, compute and study the decomposition of PGL(4) into (semialgebraic?) cells depending on the combinatorial type of the tropicalization.

# Opening:

• Let us start with P = d points in P^1 first.
• Compute the tropicalization inside P^1 x PGL(2), which gives us a 3-dim picture
• Generically, the fibers of the projections onto PGL(2) is finite
• Study the positive dimensional fibers
• Try to resolve them using modifications