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We give a necessary condition for algebraicity of finite modules over thering of formal power series. This condition is given in terms of local zeroestimates. In fact we show that this condition is also sufficient when themodule is a ring with some additional properties. To prove this result we showan effective Weierstrass Division Theorem and an effective solution to theIdeal Membership Problem in rings of algebraic power series. Finally we applythese results to prove a gap theorem for power series which are remainders ofthe Grauert-Hironaka-Galligo Division Theorem.

Local zero estimates and effective division in rings of algebraic power series