TY - JOUR
T1 - Simple birational extensions of the polynomial algebra ℂ [3]
AU - Kaliman, Shulim
AU - Vénéreau, Stéphane
AU - Zaidenberg, Mikhail
N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.
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
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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 -