On Epimorphism and related problems for linear hypersurfaces

Linear hypersurfaces over a field k have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two questions on linear polynomials of the form\\ H: = (X₁, , Xₘ) Y - F (X₁, , Xₘ, Z, T) D: =kX₁, , Xₘ, Y, Z, T: (i) Whether the affine variety V A^m+3ₖ defined by H is isomorphic to A^m+2ₖ. (ii) If V is isomorphic to an affine space, then whether H is a coordinate in D. In adv2, the first two authors had addressed these questions when is a monomial of the form (X₁, , Xₘ) = X₁^r₁ Xₘ^rₘ; rᵢ>1, \, 1 i m and F is of a certain type. In this paper, using K-theory and Gₐ-actions, we address these questions for a wider family of linear varieties. In particular, we show that when the characteristic of k is zero, F kZ, T and H defines a hyperplane (i. e. , the affine variety V defined by H is an affine space), then H is a coordinate in D along with X₁, X₂, , Xₘ. As a consequence we obtain a certain families of higher dimensional linear hyperplanes satisfying the Abhyankar-Sathaye conjecture on the Epimorphism Problem. Our results in arbitrary characteristic yield counter examples to the Zariski Cancellation Problem in positive characteristic.