## Abstract

The author uses a homogenization technique to analyze a reflection principle in complex analysis. Given a holomorphic polynomial on complex Euclidean space, the author introduces its reflection, which is a matrix of anti-holomorphic polynomials, and establishes basic properties. The introduction to the paper discusses the classical one variable case, where the reflection is well known. The main result shows that the X-variety, defined by Forstneric in his study of proper mappings between balls, is an affine space determined by the null space of the reflection matrix. An integral formula for this matrix is also given. The author provides examples both when the X-variety is a bundle and when it is not a bundle, and then proves several general results to this effect. The final result gives a new family of examples where the X-variety of a map equals the graph of the map.

Original language | English (US) |
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Pages (from-to) | 1113-1133 |

Number of pages | 21 |

Journal | Indiana University Mathematics Journal |

Volume | 52 |

Issue number | 5 |

DOIs | |

State | Published - 2003 |

## Keywords

- Proper holomorphic mappings
- Reflection principle
- Several complex variables
- Unit ball

## ASJC Scopus subject areas

- General Mathematics