Poisson brackets and twogenerated subalgebras of rings of polynomials
Authors:
Ivan P. Shestakov and Ualbai U. Umirbaev
Journal:
J. Amer. Math. Soc. 17 (2004), 181196
MSC (2000):
Primary 13F20, 13P10; Secondary 14R10, 14R15, 17B63
Published electronically:
October 3, 2003
MathSciNet review:
2015333
Fulltext PDF Free Access
Abstract 
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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 socalled 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 twogenerated 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:
http://dx.doi.org/10.1090/S0894034703004387
PII:
S 08940347(03)004387
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. 00/068328
Article copyright:
© Copyright 2003
American Mathematical Society
