Poisson brackets and two-generated subalgebras of rings of polynomials

Abstract

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.