Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

A few years ago Grigoriev introduced the concept of tropicalization of differential equations, and then Aroca et al. proved that there is an analogue of the fundamental theorem of tropical geometry, where any solution at the tropical level can be lifted to a classical solution. Unfortunately, this theory is limited in its applicability, since it only works with trivial valuations. In this talk I will describe the work of Stefano Mereta to build a general framework for tropical differential equations over non-trivially valued rings, thus allowing potential applications to p-adic differential equations. This involves some interesting twists, such as a tropical variant of the Leibniz rule. We support our theory with a differential analogue of Payne's inverse limit theorem.