Kernel of triangular derivation of the ring of polynomial of rank 3

Abstract

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 $D=\alpha x_2^l x_3^m \partial_1+\beta x_3^n \partial_2+\gamma \partial_3,$ where $\alpha ,\beta ,\gamma \in k$, 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 $k[x_1,x_2,\ldots,x_n]$ in variables $x_1,x_2,\ldots,x_n$ 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.