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Transactions of the American Mathematical Society

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

Authors: Shulim Kaliman, Stéphane Vénéreau and Mikhail Zaidenberg
Journal: Trans. Amer. Math. Soc. 356 (2004), 509-555
MSC (2000): Primary 14R10, 14R25
Published electronically: September 22, 2003
MathSciNet review: 2022709
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Abstract: The Abhyankar-Sathaye Problem asks whether any biregular embedding $\varphi:\mathbb{C}^k\hookrightarrow\mathbb{C}^n$ can be rectified, that is, whether there exists an automorphism $\alpha\in{\operatorname{Aut}}\,\mathbb{C}^n$ such that $\alpha\circ\varphi$ is a linear embedding. Here we study this problem for the embeddings $\varphi:\mathbb{C}^3\hookrightarrow \mathbb{C}^4$ whose image $X=\varphi(\mathbb{C}^3)$ is given in $\mathbb{C}^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in\mathbb{C}[x,y]\backslash\{0\}$ and $g\in\mathbb{C}[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $\mathbb{C}^{[4]}=\mathbb{C}[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $\mathbb{C}^4$). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings $\mathbb{C}^2\hookrightarrow\mathbb{C}^3$. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial $p$ as above, a criterion for when $X=p^{-1}(0)\simeq\mathbb{C}^3$.

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Additional Information

Shulim Kaliman
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124

Stéphane Vénéreau
Affiliation: Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cédex, France

Mikhail Zaidenberg
Affiliation: Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St. Martin d’Hères cédex, France

Keywords: Affine space, polynomial ring, variable, affine modification, birational extension.
Received by editor(s): December 5, 2001
Published electronically: September 22, 2003
Additional Notes: The research of the first author was partially supported by the NSA grant MDA904-00-1-0016
The third author is grateful to the IHES and to the MPI at Bonn (where a part of the work was done) for their hospitality and excellent working conditions
Article copyright: © Copyright 2003 American Mathematical Society

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