New Stably Tame Automorphisms of Polynomial Algebras

Let K[X, Y] = K[x1,…,xn, y1,…,ym] be the polynomial algebra in m + n variables over a field K of characteristic 0. Let Δ be a locally nilpotent derivation of K[X, Y] such that Δ(yi) = 0, i = 1,…,m, and let δ act as a K[Y]-affine transformation over the free K[Y]-module freely generated by x1,…,xn. We prove that the automorphism exp(wδ) of K[X, Y] is stably tame for every polynomial w from the kernel ker(Δ) of δ. This result is applied to the automorphisms of the polynomial algebra in five variables introduced recently by Drensky and Gupta and arising from wild automorphisms of generic matrix algebras. We also give an algorithm for constructing new stably tame automorphisms in any number of variables and, hence, new potential candidates for wild automorphisms.