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.

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a ‘complete’ answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on D n D^n -finiteness of solutions of first order differential equations.

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This paper presents the relationship between differential algebra and tropical differential algebraic geometry, mostly focusing on the existence problem of formal power series solutions for systems of polynomial ODE and PDE. Moreover, it improves an approximation theorem involved in the proof of the fundamental theorem of tropical differential algebraic geometry which permits to improve this latter by dropping the base field uncountability hypothesis used in the original version.

The problem of algebraic dependence of solutions to (nonlinear) first order autonomous equations over an algebraically closed field of characteristic zero is given a 'complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingredient. Classification and formal solutions of autonomous equations are treated. The results are applied to answer a question on D n -finiteness of solutions of first order differential equations. du dz . The theory of Galois groupoids does not seem to shed much light on these equations. The following example of M. Rosenlicht [15] shows that non-linear differential equations are rather different from linear ones. It