  The purpose of this paper is fourfold. The first is to develop the theory of
tropical differential algebraic geometry from scratch; the second is to present
the tropical fundamental theorem for differential algebraic geometry, and show
how it may be used to extract combinatorial information about the set of power
series solutions to a given system of differential equations, both in the
archimedean (complex analytic) and in the non-archimedean (e.g., $p$-adic)
settings. A third and subsidiary aim is to show how tropical differential
algebraic geometry is a natural application of semiring theory, and in so
doing, contribute to the valuative study of differential algebraic geometry. We
use this formalism to extend the fundamental theorem of partial differential
algebraic geometry to the differential fraction field of the ring of formal
power series in arbitrarily (finitely) many variables; in doing so we produce
new examples of non-Krull valuations that merit further study in their own
right.
