## 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.## References

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## Bibliographic 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
- Email: shestak@ime.usp.br
**Ualbai U. Umirbaev**- Affiliation: Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan
- Email: umirbaev@yahoo.com
- 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 - © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**17**(2004), 181-196 - MSC (2000): Primary 13F20, 13P10; Secondary 14R10, 14R15, 17B63
- DOI: https://doi.org/10.1090/S0894-0347-03-00438-7
- MathSciNet review: 2015333