The Nagata automorphism is wild

Abstract

It is proved that the well known Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild, that is, it can not be decomposed into a product of elementary automorphisms.

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 .…”

“…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.…”

Defining relations of the tame automorphism group of polynomial algebras in three variables

“…It was recently proved in [13,14,15,16] that the well-known Nagata automorphism (see [12]) of the polynomial algebra F [x, y, z] over a field F of characteristic 0 is wild.…”

Untitled

“…, d n ) . For example, Shestakov and Umirbaev gave in [3] an example of F ∈ Tame C 3 with mdeg F = (6, 8, 9) . See also Proposition 5.2 below.…”