Kernel of triangular derivation of the ring of polynomial of rank 3 - Abutalipova Шынар (Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series, 2024)
Let k[x_1,x_2,x_3] be an algebra of polynomials in variables x_1,x_2,x_3 over an arbitrary field k of characteristic 0 In this paper we consider triangular derivations of the form D=α x_2^l x_3^m _1+β x_3^n _2+γ _3, where α ,β ,γ k, of the algebra k[x_1,x_2,x_3]. It is well known that triangular derivations of the algebra k[x_1,x_2,x_3] are locally nilpotent. The algorithm of A. van den Essen for computing the kernel of locally nilpotent derivation of the polynomial algebra k[x_1,x_2,…,x_n] in variables x_1,x_2,…,x_n over a field k of characteristic 0 uses the map of J. Dixmier. M. Mayanishi’s theorem states that the kernel of locally nilpotent derivation of the algebra of polynomials in three variables over the field of characteristic 0 is the algebra of polynomials in two variables. In this paper, a completely new algorithm for computing the kernel of triangular derivation of the algebra of polynomials of rank 3 over a field of characteristic 0 is constructed.