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Abstrakty
Let k be a field of characteristic zero. In this note, for given linear forms L1, L2 ∈ k[x1,…, xn] and given r, s ∈ N+ = N \ {0}, we consider the equation [Lr1, P1] =[P2, Ls2] with unknowns P1, P2 ∈ k[x1, . . . , xn], and give a complete description of the set of all solutions of such an equation. Equivalently, the above equation can be written as anequation for differential forms: d(Lr1) ∧ dP1 = dP2 ∧ d(Ls2).
Słowa kluczowe
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Rocznik
Tom
Strony
11--20
Opis fizyczny
Bibliogr. 7 poz..
Twórcy
autor
- Faculty of Applied Mathematics, AGH University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Kraków, Poland
autor
- Faculty of Applied Mathematics, AGH University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Kraków, Poland
Bibliografia
- [1] A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Birkhäuser, Basel, 2000.
- [2] G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, Springer, Berlin, 2006.
- [3] D. Holik and M. Karaś, Dependence of homogeneous components of polynomials with small degree of Poisson bracket, arXiv:2101.09481v1 (2021).
- [4] A. Mostowski and M. Stark, Introduction to Higher Algebra, Pergamon Press, Oxford, 1964.
- [5] A. Nowicki, On the Jacobian equation J(f, g) = 0 for polynomials in k[x, y], Nagoya Math. J. 109 (1988), 151-157.
- [6] I. P. Shestakov and U. U. Umirbaev, The Nagata automorphism is wild, Proc. Nat. Acad. Sci. USA 100 (2003), 12561-12563.
- [7] I. P. Shestakov and U. U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials, J. Amer. Math. Soc. 17 (2004), 181-196.
Typ dokumentu
Bibliografia
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