The convergence Newton polygon of a $p$-adic differential equation I : Affinoid domains of the Berkovich affine line

Abstract

We prove that the radii of convergence of the solutions of a $p$-adic differential equation $\mathcal{F}$ over an affinoid domain $X$ of the Berkovich affine line are continuous functions on $X$ that factorize through the retraction of $X\to\Gamma$ of $X$ onto a finite graph $\Gamma\subseteq X$. We also prove their super-harmonicity properties. Roughly speaking, this finiteness result means that the behavior of the radii as functions on $X$ is controlled by a finite family of data.