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

Abstract

A characterization of tame automorphisms of the algebra A = F [ x 1 , x 2 , x 3 ] A=F[x_1,x_2,x_3] of polynomials in three variables over a field F F of characteristic 0 0 is obtained. In particular, it is proved that the well-known Nagata automorphism is wild. It is also proved that the tame and the wild automorphisms of A A are algorithmically recognizable.

The strong Nagata conjecture

“…(17) affirmatively. On the other hand, if the answer is no for some coordinate q of P 3 , then we would obtain a new proof of the Nagata conjecture without using the previous results of Shestakov and Umirbaev (9)(10)(11).…”

“…This condition can be effectively determined by an algorithm motivated by ideas of Shestakov and Umirbaev (9)(10)(11). By the algorithm, we are able to prove that all wild coordinates of K[z] [x, y] (1, 2) are also wild coordinates of P 3 ϭ K[x, y, z], hence we obtain many wild coordinates of P 3 .…”

“…Then the proof of theorem 1 in ref. 9 shows that ( f, g, h) is either elementarily reducible, or admits a reduction of one of the types I-IV.…”

“…The elementary reducibility of ( f, g, h) is algorithmically recognizable by Theorem 1, if (f, g, h) admits a reduction of one of types I-IV, then the proof is similar to the proof of theorem 3 in ref. 9.…”

“…Nagata (8) conjectured in 1972 that there exist wild automorphisms in Aut K P 3 . The famous Nagata conjecture was recently proved by Shestakov and Umirbaev (9)(10)(11). However, the following strong version of the Nagata conjecture remains open.…”