Infinite matroids in tropical differential algebra
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| Format: | Preprint |
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2023
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| author | Aroca, F. Bossinger, L. Falkensteiner, S. Lopez, C. Garay Gonzalez-Ramirez, L. R. Negrete, C. V. Valencia |
| author_facet | Aroca, F. Bossinger, L. Falkensteiner, S. Lopez, C. Garay Gonzalez-Ramirez, L. R. Negrete, C. V. Valencia |
| contents | 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 $trop(Sol(Σ))\subset (2^{\mathbb{N}^{m}})^n$ of spaces of formal power series solutions $\text{Sol}(Σ)$ of systems of linear differential equations $Σ$ in differential variables $x_1,\ldots,x_n$ having coefficients in the ring ${K}[\![t_1,\ldots,t_m]\!]$. If $Σ$ is of differential type zero, then the set $C(Sol(Σ))\subset (2^{\mathbb{N}^{m}})^n$ 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 $ϕ\circ trop(Sol(Σ))$ 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 $ϕ\circ trop(Sol(Σ))$ may not form a matroid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_04784 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Infinite matroids in tropical differential algebra Aroca, F. Bossinger, L. Falkensteiner, S. Lopez, C. Garay Gonzalez-Ramirez, L. R. Negrete, C. V. Valencia Algebraic Geometry Symbolic Computation Combinatorics 14T99, 34A30, 13N99, 05B35 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 $trop(Sol(Σ))\subset (2^{\mathbb{N}^{m}})^n$ of spaces of formal power series solutions $\text{Sol}(Σ)$ of systems of linear differential equations $Σ$ in differential variables $x_1,\ldots,x_n$ having coefficients in the ring ${K}[\![t_1,\ldots,t_m]\!]$. If $Σ$ is of differential type zero, then the set $C(Sol(Σ))\subset (2^{\mathbb{N}^{m}})^n$ 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 $ϕ\circ trop(Sol(Σ))$ 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 $ϕ\circ trop(Sol(Σ))$ may not form a matroid. |
| title | Infinite matroids in tropical differential algebra |
| topic | Algebraic Geometry Symbolic Computation Combinatorics 14T99, 34A30, 13N99, 05B35 |
| url | https://arxiv.org/abs/2305.04784 |