Linear nested Artin approximation theorem for algebraic power series

TL;DR

This paper presents a new elementary proof of the nested Artin approximation theorem for linear equations with algebraic power series, clarifying its relation to ideal operations in Noetherian local rings.

Contribution

It provides a simplified proof of the nested Artin approximation theorem and explores its connection to ideal completion and elimination in algebraic power series rings.

Findings

Elementary proof of the nested Artin approximation theorem.

Clarification of the relationship between ideal completion and elimination.

Insights into algebraic power series and Noetherian local rings.

Abstract

We give a new and elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the com-mutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals.