Stably co-tame polynomial automorphisms over commutative rings

We say that a polynomial automorphism $\varphi$ in $n$ variables is stably co-tame if the tame subgroup in $n$ variables is contained in the subgroup generated by $\varphi$ and affine automorphisms in $n+1$ variables. In this paper, we give conditions for stably co-tameness of polynomial automorphisms.