  We give a necessary condition for algebraicity of finite modules over the
ring of formal power series. This condition is given in terms of local zero
estimates. In fact we show that this condition is also sufficient when the
module is a ring with some additional properties. To prove this result 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 remainders of
the Grauert-Hironaka-Galligo Division Theorem.
