# Weierstrass polynomial

###### Definition.

A function $W:{\u2102}^{n}\to \u2102$ of the form

$$W({z}_{1},\mathrm{\dots},{z}_{n})={z}_{n}^{m}+\sum _{j=1}^{m-1}{a}_{j}({z}_{1},\mathrm{\dots},{z}_{n-1}){z}_{n}^{j},$$ |

where the ${a}_{j}$ are holomorphic functions^{} in a neighbourhood of the origin, which vanish at the origin,
is called a Weierstrass polynomial.

Any codimension 1 complex analytic subvariety of ${\u2102}^{n}$ can be written as the zero set of a Weierstrass polynomial using the Weierstrass preparation theorem. This in general cannot be done for higher codimension.

## References

- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | Weierstrass polynomial |
---|---|

Canonical name | WeierstrassPolynomial |

Date of creation | 2013-03-22 15:04:25 |

Last modified on | 2013-03-22 15:04:25 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 7 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32A17 |

Classification | msc 32B05 |

Synonym | W-polynomial |

Related topic | Multifunction |

Related topic | WeierstrassPreparationTheorem |