RESEARCH ARTICLE

On weakly semisimple derivations of the polynomial ring in two variables

Volodimir Gavran^a, Vitaliy Stepukh<sup>b

a</sup> Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str, 3, 01601, Kyiv, Ukraine
^b National Taras Shevchenko University of Kyiv, Faculty of Mechanics and Mathematics, 64, Volodymyrska str. 01033, Kyiv, Ukraine

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 $f_{1}(t), f_{2}(t)\in \mathbb K[t]$.

Keywords: polynomial ring, irreducible polynomial, Jacobian derivation.

MSC: Primary 13N15; Secondary 13N99

Received: 23.03.2014
Revised: 23.03.2014

Language: English

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