The tame and the wild automorphisms of polynomial rings in three variables
The tame and the wild automorphisms of polynomial rings in three variables
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DOI：
10.1090/s0894-0347-03-00440-5
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发表时间：
2003-10
影响因子：
3.9
通讯作者：
I. Shestakov;U. Umirbaev
中科院分区：
文献类型：
--
作者：
I. Shestakov;U. Umirbaev
设C = F [x1，x2，. . .，xn]是变量x1，x2，.. . .，xn，设AutC是域F上C作为代数的自同构群.一个自同构τ ∈ AutC称为初等的，如果它有一个形式τ：（x1，. . .，xi-1，xi，xi+1，. . .，xn）7→（x1，. . .，xi−1，αxi + f，xi+1，. . .，xn），其中0 6= α ∈ F，f ∈ F [x1，. . .，xi-1，xi+1，. . .，xn]。AutC的所有初等自同构所生成的子群称为驯服子群，来自该子群的元素称为C的驯服自同构。代数C的非驯服的自同构称为野生的。众所周知[6]、[9]、[10]、[11]，二元多项式环和自由结合代数的自同构是驯服的。目前，这些结果的一些新的证明已被发现（见[5]，[8]）。然而，在三个或更多变量的情况下，类似的问题是开放的，并被称为“代沟问题”[2]，[3]或“驯服发电机问题”[8]。一般认为答案是否定的，并且有几个候选反例（见[5]，[8]，[12]，[7]，[19]）。其中最著名的是以下自同构σ ∈ Aut（F [x，y，z]），由Nagata在1972年构造（见[12]）：σ（x）= x+（x-yz）z，σ（y）= y + 2（x-yz）x+（x-yz）z，σ（z）= z。
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.