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
Simple birational extensions of the polynomial algebra ℂ [3] . / Kaliman, Shulim; Vénéreau, Stéphane; Zaidenberg, Mikhail.
In: Transactions of the American Mathematical Society, Vol. 356, No. 2, 02.2004, p. 509-555.Research output: Contribution to journal › Article
}
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 -