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Algebra and Discrete Mathematics, 2014, Volume 18, Issue 1, Pages 50–58
(Mi adm481)
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RESEARCH ARTICLE
On weakly semisimple derivations of the polynomial ring in two variables
Volodimir Gavrana, Vitaliy Stepukhb a 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.
Received: 23.03.2014 Revised: 23.03.2014
Citation:
Volodimir Gavran, Vitaliy Stepukh, “On weakly semisimple derivations of the polynomial ring in two variables”, Algebra Discrete Math., 18:1 (2014), 50–58
Linking options:
https://www.mathnet.ru/eng/adm481 https://www.mathnet.ru/eng/adm/v18/i1/p50
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Abstract page: | 178 | Full-text PDF : | 102 | References: | 39 |
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