Infinite matroids in tropical differential algebra

Abstract

We consider a finite-dimensional vector space $W\subset K^E$ over an arbitrary field $K$ and an arbitrary set $E$. We show that the set $C(W)\subset 2^E$ consisting of the minimal supports of $W$ are the circuits of a matroid on $E$. In particular, we show that this matroid is cofinitary (hence, tame). When the cardinality of $K$ is large enough (with respect to the cardinality of $E$), then the set $trop(W)\subset 2^E$ consisting of all the supports of $W$ is a matroid itself. Afterwards we apply these results to tropical differential algebraic geometry and study the set of supports $trop(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$ of spaces of formal power series solutions $\text{Sol}(\Sigma)$ of systems of linear differential equations $\Sigma$ in differential variables $x_1,\ldots,x_n$ having coefficients in the ring ${K}[\![t_1,\ldots,t_m]\!]$. If $\Sigma$ is of differential type zero, then the set $C(Sol(\Sigma))\subset (2^{\mathbb{N}^{m}})^n$ of minimal supports defines a matroid on $E=[n]\times\mathbb{N}^{m}$, and if the cardinality of $K$ is large enough, then the set of supports $\phi \circ trop(Sol(\Sigma))$ itself is a matroid on $E$ as well. By applying the fundamental theorem of tropical differential algebraic geometry (fttdag), we give a necessary condition under which the set of solutions $Sol(U)$ of a system $U$ of tropical linear differential equations to be a matroid. We also give a counterexample to the fttdag for systems $\Sigma$ of linear differential equations over countable fields. In this case, the set $\phi \circ trop(Sol(\Sigma))$ may not form a matroid.