### 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) |
---|---|

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 2004 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{[3]}

*Transactions of the American Mathematical Society*,

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

**Simple birational extensions of the polynomial algebra ℂ ^{[3]} .** / Kaliman, Shulim; Vénéreau, Stéphane; Zaidenberg, Mikhail.

Research output: Contribution to journal › Article

^{[3]}',

*Transactions of the American Mathematical Society*, vol. 356, no. 2, pp. 509-555. https://doi.org/10.1090/S0002-9947-03-03398-1

^{[3]}Transactions of the American Mathematical Society. 2004 Feb;356(2):509-555. https://doi.org/10.1090/S0002-9947-03-03398-1

}

TY - JOUR

T1 - Simple birational extensions of the polynomial algebra ℂ [3]

AU - Kaliman, Shulim

AU - Vénéreau, Stéphane

AU - Zaidenberg, Mikhail

PY - 2004/2

Y1 - 2004/2

N2 - 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.

AB - 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.

KW - Affine modification

KW - Affine space

KW - Birational extension

KW - Polynomial ring

KW - Variable

UR - http://www.scopus.com/inward/record.url?scp=0742306234&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0742306234&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-03-03398-1

DO - 10.1090/S0002-9947-03-03398-1

M3 - Article

AN - SCOPUS:0742306234

VL - 356

SP - 509

EP - 555

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -