Geodesic
Hyperplanes of the Form f1(x, y)z 1 + · · · + fk (x, y)zk + g(x, y) Are Variables
Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 622-635

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DOI

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.
DOI : 10.4153/CMB-2005-058-7
Mots-clés : 14R10, 14R25, variables, Abhyankar–Sathaye Embedding Problem 
Vénéreau, Stéphane. Hyperplanes of the Form f1(x, y)z 1 + · · · + fk (x, y)zk + g(x, y) Are Variables. Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 622-635. doi: 10.4153/CMB-2005-058-7
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     title = {Hyperplanes of the {Form} f1(x, y)z 1 + {\textperiodcentered} {\textperiodcentered} {\textperiodcentered} + fk (x, y)zk + g(x, y) {Are} {Variables}},
     journal = {Canadian mathematical bulletin},
     pages = {622--635},
     year = {2005},
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     number = {4},
     doi = {10.4153/CMB-2005-058-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2005-058-7/}
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