In the next part of the book, which begins with Chapter 8, we will use the results from the previous chapters to make a detailed analysis of ordinary differential equations over nonarchimedean fields of characteristic 0, the motivating case being that of positive residual characteristic. However, before doing so it may be helpful to demonstrate how the results apply in a somewhat simpler setting.

In this chapter, we reconstruct some of the traditional Fuchsian theory of regular singular points of meromorphic differential equations. (The treatment is modeled on [80, §3].) We first introduce a quantitative measure of the irregularity of a singular point. We then recall how, in the case of a regular singularity (i.e., a singularity with irregularity equal to zero), one has an algebraic interpretation, using the notion of exponents, of the eigenvalues of the monodromy operator around the singular point. We then describe how to compute formal solutions of meromorphic differential equations and go on to sketch the proof of Fuchs's theorem, that the formal solutions of a regular meromorphic differential equation actually converge in some disc. We finally establish the Turrittin–Levelt–Hukuhara decomposition theorem, which gives a decomposition of an arbitrary formal differential module that is analogous to the eigenspace decomposition of a complex linear transformation. The search for an appropriate p-adic analogue of this result will lead us in Part V to the p-adic local monodromy theorem.