Extensions birationnelles simples de l'anneau de polynêmes C[3]

Translated title of the contribution: Simple birational extensions of the polynomial ring C[3]

Shulim Kaliman, Stéphane Vénéreau, Mikhail Zaidenberg

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


The Abhyankar-Sathaye problem asks whether any biregular embedding φ:CkCn can be rectified, that is, whether there exists an automorphism α∈AutCn such that αφ is a linear embedding. In the spirit of [5], here we study this problem for the embeddings φ:C3C4 whose image X=φ(C3) is given in C4 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 C4). For that, we first study the acyclicity of X in a more general setting. Then we give several equivalent conditions for X=p-1(0)≃C3 generalizing in particular a theorem of Miyanishi [4, Thm. 2].

Translated title of the contributionSimple birational extensions of the polynomial ring C[3]
Original languageFrench
Pages (from-to)319-322
Number of pages4
JournalComptes Rendus de l'Academie des Sciences - Series I: Mathematics
Issue number4
StatePublished - Aug 15 2001

ASJC Scopus subject areas

  • Mathematics(all)


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