from Part III - p-adic Differential Equations on Discs and Annuli
Published online by Cambridge University Press: 06 August 2022
In this chapter, we apply the tools developed in the preceding chapters to study the variation of the generic radius of convergence and the subsidiary radii associated to a differential module on a disc or annulus. We have already seen some instances where this study is needed to deduce consequences about convergence of solutions of p-adic differential equations. The statements we 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 a Gauss norm on the power series ring. The guiding principle is that in the visible spectrum, one should be able to relate variation of subsidiary radii to 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, 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 implement the guiding principle in a somewhat more robust manner, using the work of Chapter 6 based on matrix inequalities.
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