Abstract: Let $I$ be an ideal of the ring of Laurent polynomials $K[x_{1}^{\pm 1}, \ldots, x_{n}^{ \pm1} ]$ with coefficients in a real-valued field $(K, \nu)$. The fundamental theorem of tropical algebraic geometry states the equality $trop(V (I)) =V (trop(I))$ between the tropicalization $trop(V (I))$ of the variety $V (I) \subset (K^\ast)^ n$ and the tropical variety $V (trop(I))$ associated to the tropicalization of the ideal I.
In this talk we show this result for a differential ideal $J$ 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 $trop(Sol(J))$ of the set of solutions $Sol(J) \subset K[[t]]^ n$ of $J$, and the set of solutions $Sol(trop(J)) \subset \mathcal{P}(\mathbb{Z}_{\geq 0} )^n$ associated to the tropicalization of the ideal $J$. We show the equality $trop(Sol(J)) = Sol(trop(J))$.