### Abstract

The Abhyankar-Sathaye Problem asks whether any biregular embedding φ : ℂ ^{k} → ℂ ^{n} can be rectified, that is, whether there exists an auto-morphism α ∈ Aut ℂ ^{n} such that α o φ is a linear embedding, Here we study this problem for the embeddings φ : ℂ ^{3} → ℂ ^{4} whose image X = φ(ℂ ^{3}) is given in ℂ ^{4} by an equation p = f(x, y)u + g(x, y, z) = 0, where f ∈ ℂ[x, y]\{0} and g ∈ ℂ[x, y, z]. Under certain additional assumptions we show that, indeed, the polynomial p is a variable of the polynomial ring ℂ ^{[4]} = ℂ[x, y, z, u] (i.e., a coordinate of a polynomial automorphism of ℂ ^{4}). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings ℂ ^{2} → ℂ ^{3}. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial p as above, a criterion for when X = p ^{-1}(0) ≃ ℂ ^{3}.

Original language | English (US) |
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Pages (from-to) | 509-555 |

Number of pages | 47 |

Journal | Transactions of the American Mathematical Society |

Volume | 356 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2004 |

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### Keywords

- Affine modification
- Affine space
- Birational extension
- Polynomial ring
- Variable

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Transactions of the American Mathematical Society*,

*356*(2), 509-555. https://doi.org/10.1090/S0002-9947-03-03398-1