The Fundamental Theorem of Tropical Differential Algebraic Geometry

摘要: Let $I$ be an ideal of the ring Laurent polynomials $K[x_1^{\pm1},\ldots,x_n^{\pm1}]$ with coefficients in a real-valued field $(K,v)$. The fundamental theorem tropical algebraic geometry states equality $\text{trop}(V(I))=V(\text{trop}(I))$ between tropicalization $\text{trop}(V(I))$ closed subscheme $V(I)\subset (K^*)^n$ and variety $V(\text{trop}(I))$ associated to $\text{trop}(I)$. In this work we prove analogous result for differential $G$ $K[[t]]\{x_1,\ldots,x_n\}$, where $K$ is uncountable algebraically characteristic zero. We define $\text{trop}(\text{Sol}(G))$ set solutions $\text{Sol}(G)\subset K[[t]]^n$ $G$, $\text{trop}(G)$. These two sets are linked by morphism $\text{trop}:\text{Sol}(G)\longrightarrow \text{Sol}(\text{trop}(G))$. We show $\text{trop}(\text{Sol}(G))=\text{Sol}(\text{trop}(G))$, answering question raised D. Grigoriev earlier year.