Radii of convergence and index for $p$-adic differential operators

We study the radii of p-adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements. As an application we prove a conjecture of P. Robba (for a certain class of operators) concerning the relation between radii of convergence and index on analytic elements. We also give an explicit factorization theorem for p-adic differential operators, based on the radii of generic convergence and the slopes of the associated Newton polygon