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

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