The tame and the wild automorphisms of polynomial rings in three variables

摘要

Let C = F[x_1, x_2, ..., x_n] be the polynomial ring in the variables x_1, x_2, ..., x_n over a field F, and let AutC be the group of automorphisms of C as an algebra over F. An automorphism T AutC is called elementary if it has a formr : {x_1, ... , x_(i-1), x_i, x_(i+1), x_(i+1), ..., x_n] where 0 ≠ α, ∈F, f ∈ F[x_1, ..., x_(i-1), x_(i+1), x_n]. The subgroup of AutC generated by all the elementary automorphisms is called the tame subgroup, and the elements from this subgroup are called tame automorphisms of C. Non-tame automorphisms of the algebra C are called wild.