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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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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
Published electronically: October 3, 2003


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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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:
  • Ualbai U. Umirbaev
  • Affiliation: Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan
  • Email:
  • 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:
  • MathSciNet review: 2015333