The authors prove that Nagata’s famous example of a polynomial automorphism of \(\mathbb{C}^3\) (or more generally of \(F^3\), where \(F\) is a field of characteristic \(0\)) is not tame, i.e. can not be obtained as a composition of linear and triangular automorphisms.
Let us recall that an automorphism \((f_1,\dots, f_n): F^n\to F^n\) is called triangular if \(f_i= x_i+ g_i\), where \(g_i\in F[x_1,\dots, x_{i-1}]\) is a polynomial, for \(i= 1,\dots, n\). Let us also recall that an automorphism \((f_1,\dots, f_n): F^n\to F^n\) is called elementary if there exists \(i_0\in \{1,\dots, n\}\) such that \(f_i= x_i\) for \(i\neq i_0\) and \(f_{i_0}=\alpha x_{i_0}+ g\), where \(\alpha\neq 0\) and \(g\in F[x_1,\dots,\widehat{x_{i_0}},\dots, x_n]\). Obviously an automorphism is tame if and only if it can be obtained as a composition of elementary automorphisms.
An automorphism \((g_1, g_2, g_3)\) is called an elementary reduction of a given automorphism \((f_1, f_2, f_3)\) if there exists an elementary automorphism \(\tau\) such that \((g_1, g_2, g_3)= \tau\circ (f_1, f_2, f_3)\) and \(\deg(g_1, g_2, g_3)= \deg g_1+ \deg g_2+ \deg g_3< \deg(f_1, f_2, f_3)\). Unfortunately there are tame automorphism that do not admit any elementary reductions. To avoid this difficulties the authors define four types of special reductions (very technical definitions). In these reductions an automorphism \(\tau\) (compare definition of an elementary reduction given above) is a composition of (two, three or four) elementary automorphisms.
In the language of these reductions the main result of the paper can be formulated as the following theorem:
Theorem 2. Let \(\Theta= (f_1, f_2, f_3)\) be a tame automorphism of the ring of polynomials \(A= F[x_1, x_2, x_3]\) over a field \(F\) of characteristic \(0\). If \(\deg\Theta> 3\), then \(\Theta\) admits either an elementary reduction or a reduction of type I–IV.
As a consequence of the theorem above the authors obtain the following corollary (in which wild means not tame):
Corollary 9. The Nagata automorphism of the polynomial ring \(F[x,y,z]\) over a field \(F\) of characteristic \(0\) is wild.
The paper is very technical and the main tools developed in the second and third sections are based on the results of the previous paper of the authors published in the same volume [J. Am. Math. Soc. 17, No. 1, 181–196 (2004; Zbl 1044.17014)]. In my opinion it is a good idea to read these two papers as a sequel.