Geodesic
Poisson brackets and two-generated subalgebras of rings of polynomials
Shestakov, Ivan 1   ; Umirbaev, Ualbai 2
1 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
2 Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan
Journal of the American Mathematical Society, Tome 17 (2004) no. 1, pp. 181-196 Cet article a éte moissonné depuis la source American Mathematical Society

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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.
DOI : 10.1090/S0894-0347-03-00438-7

Shestakov, Ivan  1   ; Umirbaev, Ualbai  2

1 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
2 Department of Mathematics, Eurasian National University, Astana, 473021, Kazakhstan 
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Shestakov, Ivan; Umirbaev, Ualbai. Poisson brackets and two-generated subalgebras of rings of polynomials. Journal of the American Mathematical Society, Tome 17 (2004) no. 1, pp. 181-196. doi: 10.1090/S0894-0347-03-00438-7

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