The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations

We prove a fundamental theorem for tropical partial differential equations analogue of the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani, which work only in the case of trivial valuation as introduced by Grigoriev, to differential equations with power series coefficients over any valued field. To do so, a crucial ingredient is the framework for tropical partial differential equations introduced by Giansiracusa and Mereta. Using this framework we also add a fourth statement to the fundamental theorem, seeing the tropicalization as the set of evaluations of points of the differential Berkovich analytification on the generators of a differential algebra for a given presentation. Lastly, as a corollary of the fundamental theorem, we have that the radius of convergence of solutions of an ordinary differential equation over a nontrivially valued field can be computed tropically.