2009/11/11 by А. П. Петравчук, Anatoliy P. Petravchuk, Petravchuk, Anatoliy P.
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions #Polynomial and algebraic computation #math.AC #math.RA #msc:13N15 #msc:17A36
paper · pdf · doi:10.48550/arxiv.0911.2073
5 pages
arxiv created 2009/11/11 · arxiv updated 2009/12/01
Let k be an arbitrary field of characteristic zero, k[x, y] be the polynomial ring and D a k-derivation of the ring k[x, y]. Recall that a nonconstant polynomial F∈ k[x, y] is said to be a Darboux polynomial of the derivation D if D(F)=λF for some polynomial λ∈ k[x, y]. We prove that any two linearly independent over the field k commuting k-derivations D1 and D2 of the ring k[x, y] either have a common Darboux polynomial, or D1=D_u1, D2=D_u2 are Jacobian derivations i.e., Di(f)=det J(ui, f) for every f∈ k[x, y], i=1, 2, where the polynomials u1, u2 satisfy the condition det J(u1, u2)=c∈ k⋆. This statement about derivations is an analogue of the known fact from Linear Algebra about common eigenvectors of pairs of commuting linear operators.