Artin Approximation

Abstract

In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of A. Płoski, he conjectured that this remains true when the ring of convergent power series is replaced by a more general kind of ring. This paper presents the state of the art on this problem and its extensions. An extended introduction is aimed at non-experts. Then we present three main aspects of the subject: the classical Artin Approximation Problem, the Strong Artin Approximation Problem and the Artin Approximation Problem with constraints. Three appendices present the algebraic material used in this paper (The Weierstrass Preparation Theorem, excellent rings and regular morphisms, étale and smooth morphisms and Henselian rings). The goal is to review most of the known results and to give a list of references as complete as possible. We do not give the proofs of all the results presented in this paper but, at least, we always try to outline the proofs and give the main arguments together with precise references.

Picard–Fuchs operators for octic arrangements, I: The case of orphans

“…A more down to earth proof can be given by an application of the nested approximation theorem, [29], [38], theorem 5.2.1. In the above situation it can be applied as follows: assume we have two algebraisations X −→ S, X −→ S , given by equations of the form F (x, t) = 0, G(x , t ) = 0.…”

Linear nested Artin approximation theorem for algebraic power series

“…Let us mention that an admissible family of rings that satisfies the Weierstrass division Theorem is necessarily a family of Henselian local rings (see for instance [DL80]). But it is still unknown if an admissible family of Henselian local rings satisfies the Weierstrass division Theorem (see for instance Remark 5.20 [Ro15]). See also [La67] for a partial result in this direction.…”

Nested Artin strong approximation property

“…showing that the answer to the previous question is negative in general. On the other hand, since Theorem 1 remains valid if we replace convergent power series by algebraic power series, the question of M. Artin is also relevant in this context and it appears that in this case the question has a positive answer as it is shown in [17,Theorem 3.7] (see also [18,Corollary 3.7], [20,Theorem 5.2.1]). Similarly, we considered below the following question.…”