Hyperplanes of the Form ${f_1(x,y)z_1+\dots+f_k(x,y)z_k+g(x,y)}$ Are Variables

http://dx.doi.org/10.4153/CMB-2005-058-7
Canad. Math. Bull. 48(2005), 622-635

  Published:2005-12-01
 Printed: Dec 2005

The Abhyankar--Sathaye Embedded Hyperplane Problem asks whe\-ther any hypersurface of $\C^n$ isomorphic to $\C^{n-1}$ is rectifiable, {\em i.e.,} equivalent to a linear hyperplane up to an automorphism of $\C^n$. Generalizing the approach adopted by Kaliman, V\'en\'ereau, and Zaidenberg which consists in using almost nothing but the acyclicity of $\C^{n-1}$, we solve this problem for hypersurfaces given by polynomials of $\C[x,y,z_1,\dots, z_k]$ as in the title.