Tropical initial degeneration for systems of algebraic differential equations

We study the notion of degeneration for affine schemes associated with systems of algebraic differential equations with coefficients in the fraction field of a multivariate formal power series ring. To do this, we use an integral structure of this field that arises as the unit ball associated with the tropical valuation, first introduced in the context of tropical differential algebra. This unit ball turns out to be a particular type of integral domain, known as Bézout domain. By applying to these systems a translation map along a vector of weights that emulates the one used in classical tropical algebraic geometry, the resulting translated systems will have coefficients in this unit ball. When the resulting quotient module over the unit ball is torsion free, then it gives rise to integral models of the original system in which every prime ideal of the unit ball defines an initial degeneration, and they can be found as a base change to the residue field of the prime ideal. In particular, the closed fibres of our integral models can be rightfully called initial degenerations, since we show that there is a bijection between maximal ideals of this unit ball and monomial orders. We use this correspondence to define initial forms of differential polynomials and initial ideals of differential ideals, and we show that they share many features of their classical analogues.