Hyperplanes of the form f1 (x, y)z1+•••+fk(x, y)zk+g(x, y) are variables

The Abhyankar-Sathaye Embedded Hyperplane Problem asks whether any hypersurface of U isomorphic to Cn-1 is rectifiable, i.e., equivalent to a linear hyperplane up to an automorphism of C-n. Generalizing the approach adopted by Kaliman, Venereau, and Zaidenberg, which consists in using almost nothing but the acyclicity of Cn-1, we solve this problem for hypersurfaces given by polynomials of C[x, y, z(1),..., z(k)] as in the title.