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.