Computing syzygies over V[X1,…,Xk], V a valuation domain

We give an algorithm for computing the V-saturation of any finitely-generated submodule of V[X1,…,Xk]m (k∈N, m∈N⁎), where V is a valuation domain. Our algorithm is based on a notion of “echelon form” which ensures its correctness. The proposed algorithm terminates when two (Hilbert) series on the quotient field and the residue field of V coincide. As application, our algorithm computes syzygies over V[X1,…,Xk].