Splitting algorithms provide a list of all the roots of a polynomial, however, most numerical implementations lack the proper framework to provide certification, i.e. a computer assisted proof that the list is correct. This article adresses this issue, regardless of the underlying splitting algorithm, with minimal computational overhead. To ensure a strict control of the computational errors that is robust enough to withstand intensive iterations, we revisit and extend the theory of disk arithmetic. We also present two results, one regarding the localization of roots and one regarding the convergence of numerical refinements using Newton's method. The assumptions of both results can be certifiably checked numerically, using disk arithmetic and IEEE 754 compliant roundings. We release a compagnon in C-code that implements those results and has been used in [MV2025] to provide a computer assisted certification of the first ever tera-scale splitting.