### Abstract

We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let . M⊂CN be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point . p∈. M, for every real-algebraic subset . S'⊂CN×CN' and every positive integer . ℓ, if . f:(CN,p)→CN' is a germ of a holomorphic map such that . Graphf∩(M×CN')⊂S', then there exists a germ of a complex-algebraic map . fℓ:(CN,p)→CN' such that . Graphfℓ∩(M×CN')⊂S' and that agrees with . f at . p up to order . ℓ. .

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

Pages (from-to) | 72-88 |

Number of pages | 17 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 98 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Algebraic map
- CR manifold
- CR orbits

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Algebraic approximation in CR geometry.** / Mir, Nordine.

Research output: Contribution to journal › Article

*Journal des Mathematiques Pures et Appliquees*, vol. 98, no. 1, pp. 72-88. https://doi.org/10.1016/j.matpur.2011.11.006

}

TY - JOUR

T1 - Algebraic approximation in CR geometry

AU - Mir, Nordine

PY - 2012/7

Y1 - 2012/7

N2 - We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let . M⊂CN be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point . p∈. M, for every real-algebraic subset . S'⊂CN×CN' and every positive integer . ℓ, if . f:(CN,p)→CN' is a germ of a holomorphic map such that . Graphf∩(M×CN')⊂S', then there exists a germ of a complex-algebraic map . fℓ:(CN,p)→CN' such that . Graphfℓ∩(M×CN')⊂S' and that agrees with . f at . p up to order . ℓ. .

AB - We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let . M⊂CN be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point . p∈. M, for every real-algebraic subset . S'⊂CN×CN' and every positive integer . ℓ, if . f:(CN,p)→CN' is a germ of a holomorphic map such that . Graphf∩(M×CN')⊂S', then there exists a germ of a complex-algebraic map . fℓ:(CN,p)→CN' such that . Graphfℓ∩(M×CN')⊂S' and that agrees with . f at . p up to order . ℓ. .

KW - Algebraic map

KW - CR manifold

KW - CR orbits

UR - http://www.scopus.com/inward/record.url?scp=84862689174&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862689174&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2011.11.006

DO - 10.1016/j.matpur.2011.11.006

M3 - Article

AN - SCOPUS:84862689174

VL - 98

SP - 72

EP - 88

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

SN - 0021-7824

IS - 1

ER -