The Fundamental Theorem of Tropical Differential Algebraic Geometry

Abstract

Let $I$ be an ideal of the ring of Laurent polynomials $K[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ 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 $V(I)\subset (K^{\ast }{)}^n$ 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 $K[[t]]\{x_1,\dots ,x_n\}$, 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 $\text{trop}:\text{Sol}(G)⟶\text{Sol}(\text{trop}(G))$. We show the equality $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering a question raised by D. Grigoriev earlier this year.

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

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• Jacobi's Bound. Jacobi's results translated in K{Ö}nig's, Egerv{á}ry's and Ritt's mathematical languages
• Convergent Hahn Series and Tropical Geometry of Higher Rank
• The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems