Poisson brackets and two-generated subalgebras of rings of polynomials

Shestakov, Ivan; Umirbaev, Ualbai

doi:10.1090/S0894-0347-03-00438-7

Poisson brackets and two-generated subalgebras of rings of polynomials
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by Ivan P. Shestakov and Ualbai U. Umirbaev

J. Amer. Math. Soc. 17 (2004), 181-196

DOI: https://doi.org/10.1090/S0894-0347-03-00438-7

Published electronically: October 3, 2003

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Abstract:

We introduce a Poisson bracket on the ring of polynomials $A=F[x_1,x_2, \ldots ,x_n]$ over a field $F$ of characteristic $0$ and apply it to the investigation of subalgebras of the algebra $A$. An analogue of the Bergman Centralizer Theorem is proved for the Poisson bracket in $A$. The main result is a lower estimate for the degrees of elements of subalgebras of $A$ 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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