The Fundamental Theorem of Tropical Differential Algebraic Geometry
| Autor: | Aroca, Fuensanta, Garay, Cristhian, Toghani, Zeinab |
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| Rok vydání: | 2015 |
| Předmět: | |
| Zdroj: | Pacific J. Math. 283 (2016) 257-270 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/pjm.2016.283.257 |
| Popis: | Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem of tropical algebraic geometry states the equality $\text{trop}(V(I))=V(\text{trop}(I))$ between the tropicalization $\text{trop}(V(I))$ of the closed subscheme $V(I)\subset (K^*)^n$ and the tropical variety $V(\text{trop}(I))$ associated to the tropicalization of the ideal $\text{trop}(I)$. In this work we prove an analogous result for a differential ideal $G$ of the ring of differential polynomials $K[[t]]\{x_1,\ldots,x_n\}$, where $K$ is an uncountable algebraically closed field of characteristic zero. We define the tropicalization $\text{trop}(\text{Sol}(G))$ of the set of solutions $\text{Sol}(G)\subset K[[t]]^n$ of $G$, and the set of solutions associated to the tropicalization of the ideal $\text{trop}(G)$. These two sets are linked by a tropicalization morphism $\text{trop}:\text{Sol}(G)\longrightarrow \text{Sol}(\text{trop}(G))$. We show the equality $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering a question raised by D. Grigoriev earlier this year. Comment: 11 pages, abstract added, simplification of proofs in Sections 6 and 7, added references for Sections 1 and 7. To appear in the Pacific Journal of Mathematics |
| Databáze: | arXiv |
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