Artin’s approximation theorems and Cauchy-Riemann geometry

Mir, Nordine

doi:10.4310/maa.2014.v21.n4.a5

Cited by 9 publications

(15 citation statements)

References 28 publications

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“…In particular, it relies heavily on Artin's approximation theorem, especially in the strong approximation form first given by Wavrik. Approximation theorems and their uses in CR geometry are an interesting topic in itself; for a very good survey on the state-of-the-art in this area, we refer the reader to Mir's survey [42].…”

Section: Theorem 4 Let M M and F Be As In Theorem 2 Suppose That G Is A Lie Group With Finitely Many Connected Components With Lie Algebrmentioning

confidence: 99%

Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations

Sala

Lamel

Reiter

2020

Arkiv För Matematik

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Section: Theorem 4 Let M M and F Be As In Theorem 2 Suppose That G Is A Lie Group With Finitely Many Connected Components With Lie Algebrmentioning

confidence: 99%

Sufficient and necessary conditions for local rigidity of CR mappings and higher order infinitesimal deformations

Sala

Lamel

Reiter

2020

Arkiv För Matematik

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“…Nevertheless in [HR11], it is proven that such a system admits an Artin function using ultraproducts methods. The survey [Mir13] is a good introduction for applications of Artin Approximation in CR geometry. This result remains true is we replace C (resp.…”

Section: Milman Proved Thatmentioning

confidence: 99%

Artin Approximation

Rond¹

2018

jsing

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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.

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“…Artin approximation property with constraints is necessary in CR Geometry (see [5] and [12]), the nested case appears in the construction of the analytic deformations of a complex analytic germ when it has an isolated singularity (see [9], [11], [19]). It is also used to prove that analytic set germs are homeomorphic to set germs (see [13]), or to polynomial germs (see [6]) having no assumption on the singular locus.…”

Section: 31]) Gave An Examplementioning

confidence: 99%