On Abhyankar-Sathaye Conjecture for a family of linear hypersurfaces in ₖ⁴

Let k be a field. In this paper we study the Abhyankar-Sathaye Epimorphism Conjecture for certain hyperplanes in ₖ⁴ defined by a polynomial of the form $a(X)Y-F(X,Z,T)$. When k=, Kaliman, V\'{e}n\'{e}reau and Zaidenberg have proved that such hyperplanes are rectifiable in ⁴_. We extend their result over any field of characteristic zero and when k is a field of arbitrary characteristic we prove the Abhyankar-Sathaye Conjecture for such hyperplanes under some conditions on the roots of $a(X)$.