Artin Approximation

Journal of Singularities
volume 17 (2018), 108-192

Received: 24 April 2018.

DOI: 10.5427/jsing.2018.17g

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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. Ploski, 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.

Artin Approximation, Formal, Convergent and Algebraic Power Series, Henselian rings, Weierstrass Preparation Theorem

Mathematical Subject Classification (2010):

00-02, 03C20, 13-02, 13B40, 13J05, 13J15, 14-02, 14B12, 14B25, 32-02, 32B05, 32B10, 11J61, 26E10, 41A58.

Author(s) information:

Guillaume Rond
Aix-Marseille Université
CNRS
Centrale Marseille, I2M
Marseille, France
email: guillaume.rond@univ-amu.fr