Abstract
We prove the finiteness of the radius of convergence function $R^{M}$ of an ultrametric linear differential equation $M$ over an affinoid domain $X$ of the Berkovich affine line. This means that there exists a finite graph $\Gamma(R^{M})\subset X$, together with a canonical retraction $\delta_{R^{M}}: X\to \Gamma(R^{M})$, such that the function $R^{M}:\X\to\mathbb{R}_{> 0}$ factorizes through $\delta_{R^{M}}$. More generally, for each $\xi\in X$, we define the convergence Newton polygon $NP^{\mathrm{conv}}(M,\xi)$ of $M$, whose first slope is the logarithm of $R^{M}(\xi)$, and the other slopes are the logarithms of the radii $R^{M}_i$ of convergence of all the Taylor solutions of $M$ at $\xi$. We prove the finiteness of all the slopes $R^{M}_i(\xi)$ of $NP^{\mathrm{conv}}(M,\xi)$, and of its partial heights $H_i^{M}(\xi)$, as functions on $X$, together with their fundamental properties. Roughly speaking this result implies that there are only a finite number of numerical invariants that one can extract from the slopes of $R^{M}_i$ and $H^{M}_i$ along the branches of $X$. As a corollary we have their continuity.