Minimal solutions of tropical linear differential systems

We introduce and study minimal (with respect to inclusion) solutions of systems of tropical linear differential equations. We describe the set of all minimal solutions for a single equation. It is shown that any tropical linear differential equation in a single unknown has either a solution or a solution at infinity. For a generic system of $n$ tropical linear differential equations in $n$ unknowns, upper and lower bounds on the number of minimal solutions are established. The upper bound involves inversions of a family of permutations which generalize inversions of a single permutation. For $n=1, 2$, we show that the bounds are sharp.