On weakly semisimple derivations of the polynomial ring in two variables

Abstract

Let \(\mathbb K\) be an algebraically closed field of characteristic zero and \(\mathbb K[x,y]\) the polynomial ring. Every element \(f\in \mathbb K[x,y]\) determines the Jacobian derivation \(D_f\) of \(\mathbb K[x,y]\) by the rule D_f(h) = det J(f,h), where J(f,h) is the Jacobian matrix of the polynomials \(f\) and \(h\). A polynomial \(f\) is called weakly semisimple if there exists a polynomial \(g\) such that \(D_f(g) = \lambda g\) for some nonzero \(\lambda\in \mathbb K\). Ten years ago, Y. Stein posed a problem of describing all weakly semisimple polynomials (such a description would characterize all two dimensional nonabelian subalgebras of the Lie algebra of all derivations of \(\mathbb K[x,y]\) with zero divergence). We give such a description for polynomials \(f\) with the separated variables, i.e. which are of the form: \(f(x,y) = f_1(x) f_2(y)\) for some \(f_{1}(t), f_{2}(t)\in \mathbb K[t]\).