Infinite matroids in tropical differential algebra
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We consider a finite-dimensional vector space W⊂ K^E over an arbitrary field K and an arbitrary set E. We show that the set C(W)⊂ 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)⊂ 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(Σ))⊂ (2^ℕ^m)^n of spaces of formal power series solutions Sol(Σ) of systems of linear differential equations Σ in differential variables x_1,…,x_n having coefficients in the ring K[[t_1,…,t_m]]. If Σ is of differential type zero, then the set C(Sol(Σ))⊂ (2^ℕ^m)^n of minimal supports defines a matroid on E=ℕ^mn, and if the cardinality of K is large enough, then the set of supports trop(Sol(Σ)) 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 Σ of linear differential equations over countable fields. In this case, the set trop(Sol(Σ)) may not form a matroid.
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