Simple birational extensions of the polynomial algebra ℂ [3]

Shulim Kaliman, Stéphane Vénéreau, Mikhail Zaidenberg

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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 languageEnglish (US)
Pages (from-to)509-555
Number of pages47
JournalTransactions of the American Mathematical Society
Issue number2
StatePublished - Feb 2004


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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics


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