On $\mathbb{A}^1$-contractibility of certain simple birational extensions of affine spaces

Over a field of characteristic zero, up to isomorphism of varieties, affine spaces are the only smooth $\mathbb{A}^1$-contractible affine varieties in dimensions $\leqslant 2$. However, in dimensions $\geqslant 3$, examples of smooth $\mathbb{A}^1$-contractible affine varieties that are not isomorphic to affine spaces have been constructed in recent works of Dubouloz--Fasel and Dubouloz--Ghosh.
In this paper, we consider a generalized class of smooth affine varieties containing these examples of $\mathbb{A}^1$-contractible varieties, given by $$ a(x_m)b(x_1,\ldots,x_{m-1})y+f(z,t)+x_m=0, $$ and investigate when such a variety is actually isomorphic to an affine space. We establish that, for large subfamilies of these varieties, being isomorphic to an affine space is equivalent to the rectifiability of the corresponding hyperplane embedding in $\mathbb{A}^{m+3}$; that is, there exists an automorphism of the ambient affine space carrying the hypersurface onto a coordinate hyperplane. Thus, our result naturally connects $\mathbb{A}^1$-contractibility of these varieties with the classical embedding problem for affine spaces in codimension one, and provides new evidence toward the Abhyankar--Sathaye conjecture. A key ingredient in our approach is the study of singular $\mathbb{A}^1$-contractible affine curves over fields of characteristic zero. We describe the possible singularities of such curves and obtain a generalization of the classical result of Lin--Zaidenberg for topologically contractible affine plane curves.