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

TL;DR

This paper establishes a necessary and sufficient condition for algebraicity of finite modules over rings of formal power series using local zero estimates, and develops effective division and ideal membership algorithms.

Contribution

It introduces local zero estimates as a criterion for algebraicity and provides effective division and ideal membership algorithms in rings of algebraic power series.

Findings

Established a necessary condition for algebraicity based on local zero estimates.

Proved an effective Weierstrass Division Theorem for algebraic power series.

Applied results to prove a gap theorem for remainders in division of power series.

Abstract

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.