### Abstract

Let M be a real-analytic CR submanifold of CN and S′ be a realanalytic subset of CN+N′ We say that the pair (M,S′) has the Artin approximation property if for every point p ∈ M and every positive integerℓif H: (CN, p) → CN′ is a formal holomorphic map such that GraphH ∩ (M × CN′ ) ⊂S′, there exists a germ at p of a holomorphic map h^{l} (CN, p) → CN′ which agrees with H at p up to order ^{l}satisfying Graph h ∩ (M × CN′ ) ⊂S′. In this paper, we give some sufficient conditions on a pair (M,S′) to have the Artin approximation property. We show that if the CR orbits of M are all of the same dimension and at most of codimension one in M and if S′ is any partially algebraic subset of CN × CN′ then (M,S′) has the Artin approximation property.

Original language | English |
---|---|

Pages (from-to) | 221-244 |

Number of pages | 24 |

Journal | Mathematical Research Letters |

Volume | 23 |

Issue number | 1 |

Publication status | Published - 2016 |

### Fingerprint

### Keywords

- Artin approximation
- CR manifold
- Formal map

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*23*(1), 221-244.

**On Artin approximation for formal CR mappings.** / Mir, Nordine.

Research output: Contribution to journal › Article

*Mathematical Research Letters*, vol. 23, no. 1, pp. 221-244.

}

TY - JOUR

T1 - On Artin approximation for formal CR mappings

AU - Mir, Nordine

PY - 2016

Y1 - 2016

N2 - Let M be a real-analytic CR submanifold of CN and S′ be a realanalytic subset of CN+N′ We say that the pair (M,S′) has the Artin approximation property if for every point p ∈ M and every positive integerℓif H: (CN, p) → CN′ is a formal holomorphic map such that GraphH ∩ (M × CN′ ) ⊂S′, there exists a germ at p of a holomorphic map hl (CN, p) → CN′ which agrees with H at p up to order lsatisfying Graph h ∩ (M × CN′ ) ⊂S′. In this paper, we give some sufficient conditions on a pair (M,S′) to have the Artin approximation property. We show that if the CR orbits of M are all of the same dimension and at most of codimension one in M and if S′ is any partially algebraic subset of CN × CN′ then (M,S′) has the Artin approximation property.

AB - Let M be a real-analytic CR submanifold of CN and S′ be a realanalytic subset of CN+N′ We say that the pair (M,S′) has the Artin approximation property if for every point p ∈ M and every positive integerℓif H: (CN, p) → CN′ is a formal holomorphic map such that GraphH ∩ (M × CN′ ) ⊂S′, there exists a germ at p of a holomorphic map hl (CN, p) → CN′ which agrees with H at p up to order lsatisfying Graph h ∩ (M × CN′ ) ⊂S′. In this paper, we give some sufficient conditions on a pair (M,S′) to have the Artin approximation property. We show that if the CR orbits of M are all of the same dimension and at most of codimension one in M and if S′ is any partially algebraic subset of CN × CN′ then (M,S′) has the Artin approximation property.

KW - Artin approximation

KW - CR manifold

KW - Formal map

UR - http://www.scopus.com/inward/record.url?scp=84973473718&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84973473718&partnerID=8YFLogxK

M3 - Article

VL - 23

SP - 221

EP - 244

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 1

ER -