Algebraic approximation in CR geometry

Mir, Nordine

doi:10.1016/j.matpur.2011.11.006

Cited by 4 publications

(9 citation statements)

References 18 publications

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“…This subvariety V ′ is however in general strictly larger than the subvariety V 1 consisting of the set of points of p ∈ M where the dimension of the CR orbits is not constant in any neighborhood of p (as defined in Section 2.2). The conclusion of Theorem 2.14 with V replaced by V 1 has been established more recently by the author in [Mir12]. The result can be stated as follows:…”

Section: Cr Submanifolds Let M ⊂ Cmentioning

confidence: 77%

“…Theorem 2.15 is a consequence of a more general result proved in [Mir12] that is developed in detail later in this article in Section 3.2. The following elementary example shows that V 1 is in general strictly contained in V .…”

Section: Cr Submanifolds Let M ⊂ Cmentioning

confidence: 80%

“…Theorem 3.9 ( [Mir12]). For every real-algebraic CR submanifold M ⊂ C N , there exists a closed proper real-algebraic subvariety Σ M of M such that M \ Σ M has the Nash-Artin approximation property.…”

Section: There Exists a Sequence Of Germs Of Algebraic Holomorphic Mamentioning

confidence: 99%

“…Theorem 3.10 ( [Mir12]). Any real-algebraic hypersurface of C N has the NashArtin approximation property.…”

Section: There Exists a Sequence Of Germs Of Algebraic Holomorphic Mamentioning

confidence: 99%

“…The first one is the use of Artin's approximation theorem [A69] on a ground field that is different from the usual fields of real and complex numbers. Indeed, one needs to apply such a result to some field extension over C defined by some ratios of formal power series (see [Mir12] for more details). The other important ingredient is the use of a version of Artin's approximation theorem (for so-called nested subrings) proved by Popescu [P86].…”

Section: There Exists a Sequence Of Germs Of Algebraic Holomorphic Mamentioning

confidence: 99%

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Artin’s approximation theorems and Cauchy-Riemann geometry

Mir

2014

Methods and Applications of Analysis

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Abstract. Artin's approximation theorems are powerful tools in analytic and algebraic geometry for finding solutions of systems of analytic or algebraic equations whenever a given formal solution exists. In this survey article we describe the recent developments involving the use of Artin's approximation theorems in some problems arising from Cauchy-Riemann geometry. The solution to such problems simultaneously lead to a number of results that can be stated as PDE versions of Artin's approximation theorems. The article is intended to a non-expert audience. A number of examples and open problems are also mentioned.

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Section: Cr Submanifolds Let M ⊂ Cmentioning

confidence: 77%

Section: Cr Submanifolds Let M ⊂ Cmentioning

confidence: 80%

Section: There Exists a Sequence Of Germs Of Algebraic Holomorphic Mamentioning

confidence: 99%

“…Theorem 3.10 ( [Mir12]). Any real-algebraic hypersurface of C N has the NashArtin approximation property.…”

Section: There Exists a Sequence Of Germs Of Algebraic Holomorphic Mamentioning

confidence: 99%

Section: There Exists a Sequence Of Germs Of Algebraic Holomorphic Mamentioning

confidence: 99%

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Artin’s approximation theorems and Cauchy-Riemann geometry

Mir

2014

Methods and Applications of Analysis

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Linear nested Artin approximation theorem for algebraic power series

2018

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We give an elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin's question about linear nested approximation problem are equivalent.

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Transcendental holomorphic maps between real algebraic manifolds in a complex space

Rond¹

2020

Proc. Amer. Math. Soc.

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We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.2010 Mathematics Subject Classification. Primary: 32H02, Secondary: 05A15, 14P05, 32C05, 32V40, 39B32.Key words and phrases. holomorphic map, algebraic map, Bishop surface, lattice walk. The author is deeply grateful to the UMI LASOL of the CNRS where this project has been carried out.

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Algebraic approximation in CR geometry

Cited by 4 publications

References 18 publications

Artin’s approximation theorems and Cauchy-Riemann geometry

Artin’s approximation theorems and Cauchy-Riemann geometry

Linear nested Artin approximation theorem for algebraic power series

Transcendental holomorphic maps between real algebraic manifolds in a complex space

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