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:=α(X₁,,Xₘ)Y - F(X₁,, Xₘ,Z,T)∈ D:=k[X₁,…,Xₘ, Y,Z,T]: (i) Whether the affine variety V∈ Aᵐ⁺³ₖ defined by H is isomorphic to Aᵐ⁺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 α is a monomial of the form α(X₁,…,Xₘ) = X₁r₁ Xₘrₘ; 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 ∈ k[Z,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.