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We prove that the radii of convergence of the solutions of a p-adic differential equation \({\fancyscript{F}}\) 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. This finiteness result means that the behavior of the radii as functions on X is controlled by a finite family of data.

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