If it is quite easy to check a given integer is a root of a given polynomial with integer coefficients, verifying we know all the integral roots of a polynomial requires a different approach. In both univariate and bivariate cases, we introduce a type of integral roots certificates and the corresponding checker specification, based on Hensel's lifting. We provide a formalization of this iterative algorithm from which we deduce a formal proof of the correctness of the checkers, with the help of the COQ proof assistant along with the SSReflect extension. The ultimate goal of this work is to provide a component that will be involved in a complete certification chain for solving the Table Maker's Dilemma in an exact way.