On the triviality of a family of linear hyperplanes

Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of ${\mathbb{A}}^{m+3}$ defined by a linear relation of the form $x_1^{r_1}\cdots x_m^{r_m}y=F(x_1,\dots ,x_m,z,t)$, $A$ the coordinate ring of $\mathbb{V}$ and $G=X_1^{r_1}\cdots X_m^{r_m}Y-F(X_1,\dots ,X_m,Z,T)$. In \cite{com}, the second author had studied the case $m=1$ and had obtained several necessary and sufficient conditions for $\mathbb{V}$ to be isomorphic to the affine 3-space and $G$ to be a coordinate in $k[X_1,Y,Z,T]$. In this paper, we study the general higher-dimensional variety $\mathbb{V}$ for each $m⩾1$ and obtain analogous conditions for $\mathbb{V}$ to be isomorphic to ${\mathbb{A}}^{m+2}$ and $G$ to be a coordinate in $k[X_1,\dots ,X_m,Y,Z,T]$, under a certain hypothesis on $F$. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral domains of the type $A$ under certain conditions. These results show that for each $d⩾3$, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension $d$ in positive characteristic.