Let I be an ideal of the ring of Laurent polynomials K[x ±1 1 , . . . , x ±1 n ] with coefficients in a real-valued field (K, v). The fundamental theorem of tropical algebraic geometry states the equality trop(V (I)) = V (trop(I)) between the tropicalization trop(V (I)) of the closed subscheme V (I) ⊂ (K * ) n and the tropical variety V (trop(I)) associated to the tropicalization of the ideal trop(I).In this work we prove an analogous result for a differential ideal G of the ring of differential polynomials K[[t]]{x1, . . . , xn}, where K is an uncountable algebraically closed field of characteristic zero. We define the tropicalization trop(Sol(G)) of the set of solutions Sol(G) ⊂ K[[t]] n of G, and the set of solutions Sol(trop(G)) ⊂ P(Z ≥0 ) n associated to the tropicalization of the ideal trop(G). These two sets are linked by a tropicalization morphism trop : Sol(G) −→ Sol(trop(G)).We show the equality trop(Sol(G)) = Sol(trop(G)), answering a question raised by D. Grigoriev earlier this year.

Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions. We show that the Fundamental Theorem can be extended to the case of systems of partial differential equations by introducing vertex sets of Newton polytopes. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic manipulation.

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Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions. We show that the Fundamental Theorem can be extended to the case of systems of partial differential equations by introducing vertex sets of Newton polygons.