Minimal solutions of tropical linear differential systems
Dima Grigoriev, Cristhian Garay López
TL;DR
The paper develops a comprehensive framework for minimal solutions of tropical linear differential systems (TLDEs). It first clarifies the univariate case, giving a complete description of minimal solutions in terms of tropical hyperplanes and a finite exceptional set, and then extends to multivariate systems where the finite part $\mathcal{C}_{\mathbb{Z}_{\ge0}}(P)$ embeds into a valuated matroid together with a finite set $F(P)$. For generic regular systems with $m=n$, sharp bounds on the number of minimal solutions are established in low dimensions ($n=1$ yields at most $k$; $n=2$ yields at most $\frac{(k_1+k_2)(k_1+k_2+1)}{2}$), while broader lower/upper bounds for $n>2$ are derived using inversions of permutation families. A key tool is the family of tropical polynomials $\{A_{\alpha,j}\}$ and the tropical discriminant, which together characterize regularity and holonomicity, linking tropical differential algebraic geometry to matroid theory and permutation combinatorics. Overall, the results provide concrete combinatorial descriptions and algorithmic criteria for the structure of tropical solution spaces and for deciding holonomicity in TLDE systems.
Abstract
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.