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

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

Published electronically:
October 3, 2003

MathSciNet review:
2015333

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Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a Poisson bracket on the ring of polynomials over a field of characteristic and apply it to the investigation of subalgebras of the algebra . An analogue of the Bergman Centralizer Theorem is proved for the Poisson bracket in . The main result is a lower estimate for the degrees of elements of subalgebras of generated by so-called -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

Email:
shestak@ime.usp.br

**Ualbai U. Umirbaev**

Affiliation:
Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan

Email:
umirbaev@yahoo.com

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

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