• DOI:10.1090/S0894-0347-03-00440-5
• Corpus ID: 33765885

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

@article{Shestakov2003TheTA,
  title={The tame and the wild automorphisms of polynomial rings in three variables},
  author={Ivan P. Shestakov and Ualbai U. Umirbaev},
  journal={Journal of the American Mathematical Society},
  year={2003},
  volume={17},
  pages={197-227}
}

• I. Shestakov, U. Umirbaev
• Published 3 October 2003
• Mathematics
• Journal of the American Mathematical Society

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.