We characterize finite modules over the ring of formal power series which are completion of modules defined over the ring of algebraic power series. This characterization is made in terms of local zero estimates. To prove this characterization we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally we apply these results to prove a gap theorem for power series which are remainder of the Grauert-Hironaka-Galligo Division Theorem.