We construct and study length 2 variables of A[x,y] (A is a commutative ring). If A is an integral domain, we determine among these variables those which are tame. If A is a UFD, we prove that these variables are all stably tame. We apply this construction to show that some polynomials of A[x₁,...,xₙ] are variables using transfer.

We show that every n-dimensional smooth algebraic variety X can be covered by Zariski open subsets $U_i$ which are isomorphic to closed smooth hypersurfaces in ${ℂ}^{n+1}$. As an application we show that forevery (pure) n-1-dimensional ℂ-uniruled variety $X\subset {ℂ}^m$ there is a generically-finite (even quasi-finite) polynomial mapping $f:{ℂ}^n\to {ℂ}^m$ such that $X\subset S_f$. This gives (together with [3]) a full characterization of irreducible components of the set $S_f$ for generically-finite polynomial mappings $f:{ℂ}^n\to {ℂ}^m$.

We give a new proof of Miyanishi's theorem on the classification of the additive group scheme actions on the affine plane.