A note on the solution set of the equation [Lr1, P1]=[P2, Ls2] for given linear forms L1, L2

• Autorzy: Holik Daria , Karaś Marek

Identyfikatory

DOI

10.4064/ba210123-12-7

• Języki publikacji: EN

Abstrakty

EN

Let k be a field of characteristic zero. In this note, for given linear forms L₁, L₂ ∈ k[x₁,…, x_n] and given r, s ∈ N₊ = N \ {0}, we consider the equation [L^r₁, P₁] =[P₂, L^s₂] with unknowns P₁, P₂ ∈ k[x₁, . . . , x_n], 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(L^r₁) ∧ dP₁ = dP₂ ∧ d(L^s₂).

Słowa kluczowe

EN

poisson bracket; polynomial ring; differential forms

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2021
• Tom: Vol. 69, no. 1
• Strony: 11--20
• Opis fizyczny: Bibliogr. 7 poz..

Twórcy

autor

Holik Daria

• Faculty of Applied Mathematics, AGH University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Kraków, Poland

• https://orcid.org/0000-0002-2133-9711

autor

Karaś Marek

• Faculty of Applied Mathematics, AGH University of Science and Technology, Al. A. Mickiewicza 30, 30-059 Kraków, Poland

• https://orcid.org/0000-0003-0821-521X

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.