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 $D:=k[X_1,\ldots,X_m,Y,Z,T]$ over an arbitrary field $k$ defined by the linear polynomials of the form:

$$H:=a(X_1,\ldots,X_m)Y-F(X_1,...,X_m,Z,T)$$

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 $X_1,X_2,\ldots,X_m$. 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.