In this chapter, we apply the tools developed in the preceding chapters to study the variation of the generic radius of convergence, and of the subsidiary radii, associated with a differential module on a disc or annulus. We have already seen some instances where this study is needed to deduce consequences about the convergence of solutions of p-adic differential equations (Examples 9.6.2 and 9.9.3).

The statements we will formulate are modeled on statements governing the variation of the Newton polygon of a polynomial over a ring of power series as we vary the choice of Gauss norm on the power series ring. The guiding principle is that, in the visible spectrum, one should be able to relate the variation of subsidiary radii to the variation of Newton polygons via matrices of action of the derivation on suitable bases. This includes the relationship between subsidiary radii and Newton polygons for cyclic vectors (Theorem 6.5.3), but trying to use that approach directly creates no end of difficulties because cyclic vectors only exist in general for differential modules over fields. We will implement the guiding principle in a somewhat more robust manner than before, using the discussion of matrix inequalities in Chapter 6.

To this principle we must add the techniques of descent along a Frobenius morphism introduced in Chapter 10, including the off-centered variant. This allows us to overcome the limitation to the visible spectrum.