Computing the V-saturation of finitely-generated submodules of V[X]m where V is a valuation domain

Recently, Lombardi, Quitté and Yengui have given a Gröbner-free algorithm which computes the V-saturation of any finitely generated submodule of V[X]n, where V is a valuation domain. The goal of this paper is to clarify this algorithm, to give precise complexity bounds, and a complete submodule membership test for the saturation. As application, we give precise degree bounds on syzygies over V[X].