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Journal Articles Journal of Singularities Year : 2018

Artin Approximation

Guillaume Rond


— 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 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, étales 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. This paper is an extended version of the habilitation thesis of the author. The author wishes to thank the members of the jury of his habilitation thesis who encouraged him to improve the first version of this writing : Edward Biers-tone, Charles Favre, Herwig Hauser, Michel Hickel, Adam Parusiński, Anne Pichon and Bernard Teissier. The author wishes to thank especially Herwig Hauser for his constant support and encouragement and for his relevant remarks on the first stages of this writing. In particular most of the examples given in the introduction were his proposal. The author also wishes to thank the participants of the Chair Jean Morlet at CIRM for the fruitful discussions that help him to improve this text, in particular the participants of the doctoral school and more specifically
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hal-01255285 , version 1 (13-01-2016)



Guillaume Rond. Artin Approximation. Journal of Singularities, 2018, 17, pp.108-192. ⟨10.5427/jsing.2018.17g⟩. ⟨hal-01255285⟩
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