On embedding of linear hypersurfaces and the Zariski Cancellation Problem

In this talk we shall give a brief overview and address some recent developments on the above two problems.We will exhibit several families of hypersurfaces in the polynomial ring D:=k[X₁,...,X_m,Y,Z,T] over an arbitrary field k defined by the linear polynomials of the form :

                        H:=a(X₁,...,X_m)Y-F(X₁,...,X_m,Z,T)

satisfying the Abhyankar–Sathaye Conjecture on the Epimorphism/Embedding Problem. For instance, we will show that when the characteristic of the field k is zero, F is a polynomial in Z and T only and H defines a hyperplane (i.e., the affine variety defined by H is an affine space), then H is a coordinate in D along with X₁,X₂,...,X_m. Our results also yield new infinite family of non-isomorphic counterexamples in positive characteristic to the Zariski Cancellation Problem.

This talk is based on joint works with Neena Gupta and Parnashree Ghosh.