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

# Group leader

The group leader for this question is Yue Ren.

Bibliography

1. D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Graduate Studies in Mathematics, 161. American Mathematical Society, Providence, RI, 2015.