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.
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.
We introduce a Poisson bracket on the ring of polynomials
A
=
F
[
x
1
,
x
2
,
…
,
x
n
]
A=F[x_1,x_2, \ldots ,x_n]
over a field
F
F
of characteristic
0
0
and apply it to the investigation of subalgebras of the algebra
A
A
. An analogue of the Bergman Centralizer Theorem is proved for the Poisson bracket in
A
A
. The main result is a lower estimate for the degrees of elements of subalgebras of
A
A
generated by so-called
∗
\ast
-reduced pairs of polynomials. The estimate involves a certain invariant of the pair which depends on the degrees of the generators and of their Poisson bracket. It yields, in particular, a new proof of the Jung theorem on the automorphisms of polynomials in two variables. Some relevant examples of two-generated subalgebras are given and some open problems are formulated.
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