## Abstract

The Vénéreau polynomials v_{n}:y+x^{n}(xz+y(yu+z^{2})), n≥1, on A_{ℂ} ^{4} have all fibers isomorphic to the affine space A_{ℂ}^{3}. Moreover, for all n≥1 the map (v_{n},x):A_{ℂ}^{4}→ A_{ℂ}^{2} yields a flat family of affine planes over A_{ℂ}^{2}. In the present note we show that over the punctured plane A_{ℂ}^{2}\{0̄}, this family is a fiber bundle. This bundle is trivial if and only if v_{n} is a variable of the ring ℂ[x][y,z,u] over ℂ[x]. It is an open question whether v_{1} and v_{2} are variables of the polynomial ring ℂ^{[4]}=ℂ[x,y,z,u], whereas Vénéreau established that v_{n} is indeed a variable of ℂ[x][y,z,u] over ℂ[x] for n≥3. In this note we give another proof of Vénéreau's result based on the above equivalence. We also discuss some other equivalent properties, as well as the relations to the Abhyankar-Sathaye Embedding Problem and to the Dolgachev-Weisfeiler Conjecture on triviality of flat families with fibers affine spaces.

Original language | English (US) |
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Pages (from-to) | 275-286 |

Number of pages | 12 |

Journal | Journal of Pure and Applied Algebra |

Volume | 192 |

Issue number | 1-3 |

DOIs | |

State | Published - Sep 1 2004 |

## ASJC Scopus subject areas

- Algebra and Number Theory