DESCRIPTION
In many applications of Artin approximation, certain components of the power series solutions are required to depend only on certain subsets of the set of all variables. If these subsets are arbitrarily distributed, there is no hope to find convergent solutions, as examples of Gabrielov and Becker demonstrate.
However, if the subsets are nested, i.e., form an ascending chain of subsets, Artin approximation holds for algebraic systems of equations, as was shown by Popescu and Spivakovsky. The proofs are very long and require difficult machinery as e.g. Néron desingularization. The proposed research program intends to clarify and simplify these proofs. Already the case of linear equations is very interesting. In the classical situation of Artin approximation, i.e., without constraints, it corresponds to the flatness of the formal power series ring over the convergent power series ring, and is proven for example by the extension by GrauertHironakaGalligo of the Weierstrass division theorem to the case of ideals. In the presence of constraints by nested subring conditions, the respective generalization of the division theorem fails already for convergent power series. However, for algebraic series, there is a good chance that the theorem can be proven, using techniques developed by AlonsoGarcia, CastroJiménez and Hauser. Once this is established, it remains to reduce the general nested subring approximation problem to the linear case. This is a linearization problem for textile maps between power series spaces as proposed by BruschekHauser. For arc spaces, i.e.,one variable, this has already been successfully applied. 
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