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Article Dans Une Revue Transactions of the American Mathematical Society Année : 2004

Simple birational extensions of the polynomial algebra $\C^{[3]}$

Résumé

The Abhyankar-Sathaye Problem asks whether any biregular embedding of affine spaces $A^m_k\to A^n_k$ can be rectified, that is, is equivalent to a linear embedding up to an automorphism of the target space. Here we study this problem for the embeddings $C^3 \to C^4$ whose image $X$ is given in $C^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in C[x,y],$ $f\neq 0$ and $g\in C[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $C[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $C^4$). This is an analog of a theorem due to Sathaye which concerns the case of embeddings $C^2\to C^3$. Besides, we generalize a theorem of Miyanishi giving, for a polynomial $p$ as above, a criterion for as when $X$ is isomorphic to $C^3$.

Dates et versions

hal-00323568 , version 1 (22-09-2008)

Identifiants

Citer

Sh. Kaliman, Mikhail Zaidenberg, Stéphane Vénéreau. Simple birational extensions of the polynomial algebra $\C^{[3]}$. Transactions of the American Mathematical Society, 2004, 356 (2), pp.509-555. ⟨10.1090/S0002-9947-03-03398-1⟩. ⟨hal-00323568⟩

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