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Hyperplanes of the Form f1(x, y)z 1 + · · · + fk (x, y)zk + g(x, y) Are Variables

Published online by Cambridge University Press:  20 November 2018

Stéphane Vénéreau
Stéphane Vénéreau*
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, switzerland e-mail: stephane.venereau@unibas.ch
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Abstract

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The Abhyankar–Sathaye Embedded Hyperplane Problem asks whether any hypersurface of ${{\mathbb{C}}^{n}}$ isomorphic to ${{\mathbb{C}}^{n-1}}$ is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of ${{\mathbb{C}}^{n}}$ . Generalizing the approach adopted by Kaliman, Vénéreau, and Zaidenberg, which consists in using almost nothing but the acyclicity of ${{\mathbb{C}}^{n-1}}$ , we solve this problem for hypersurfaces given by polynomials of $\mathbb{C}\left[ x,y,{{z}_{1}},...,{{z}_{k}} \right]$ as in the title.

Keywords

14R1014R25variablesAbhyankar–Sathaye Embedding Problem

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 622 - 635
Copyright
Copyright © Canadian Mathematical Society 2005

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