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Poisson brackets and two-generated subalgebras of rings of polynomials

Authors: Ivan P. Shestakov and Ualbai U. Umirbaev
Journal: J. Amer. Math. Soc. 17 (2004), 181-196
MSC (2000): Primary 13F20, 13P10; Secondary 14R10, 14R15, 17B63
Published electronically: October 3, 2003
MathSciNet review: 2015333
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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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Additional Information

Ivan P. Shestakov
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo - SP, 05311–970, Brazil; Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
MR Author ID: 289548

Ualbai U. Umirbaev
Affiliation: Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan

Keywords: Rings of polynomials, Poisson brackets, subalgebras
Received by editor(s): January 8, 2003
Published electronically: October 3, 2003
Additional Notes: The first author was supported by CNPq
The second author was supported by the FAPESP Proc. 00/06832-8
Article copyright: © Copyright 2003 American Mathematical Society