The Tame and the Wild Automorphisms of Polynomial Rings in Three Variables

Let C = F [x1, x2, . . . , xn] be the polynomial ring in the variables x1, x2, . . . , xn over a field F , and let AutC be the group of automorphisms of C as an algebra over F . An automorphism τ ∈ AutC is called elementary if it has a form τ : (x1, . . . , xi−1, xi, xi+1, . . . , xn) 7→ (x1, . . . , xi−1, αxi + f, xi+1, . . . , xn), where 0 6= α ∈ F, f ∈ F [x1, . . . , xi−1, xi+1, . . . , xn]. 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. It is well known [6], [9], [10], [11] that the automorphisms of polynomial rings and free associative algebras in two variables are tame. At present, a few new proofs of these results have been found (see [5], [8]). However, in the case of three or more variables the similar question was open and known as “The generation gap problem” [2], [3] or “Tame generators problem” [8]. The general belief was that the answer is negative, and there were several candidate counterexamples (see [5], [8], [12], [7], [19]). The best known of them is the following automorphism σ ∈ Aut(F [x, y, z]), constructed by Nagata in 1972 (see [12]): σ(x) = x+ (x − yz)z, σ(y) = y + 2(x − yz)x+ (x − yz)z, σ(z) = z. Observe that the Nagata automorphism is stably tame [17]; that is, it becomes tame after adding new variables. The purpose of the present work is to give a negative answer to the above question. Our main result states that the tame automorphisms of the polynomial ring A = F [x, y, z] over a field F of characteristic 0 are algorithmically recognizable. In particular, the Nagata automorphism σ is wild. The approach we use is different from the traditional ones. The novelty consists of the imbedding of the polynomial ring A into the free Poisson algebra (or the