On weakly semisimple derivations of the polynomial ring in two variables
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 pol...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1046 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-10462018-04-26T02:28:53Z On weakly semisimple derivations of the polynomial ring in two variables Gavran, Volodimir Stepukh, Vitaliy polynomial ring, irreducible polynomial, Jacobian derivation 13N15; 13N99 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 \(f_{1}(t), f_{2}(t)\in \mathbb K[t]\). Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1046 Algebra and Discrete Mathematics; Vol 18, No 1 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1046/568 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
polynomial ring irreducible polynomial Jacobian derivation 13N15; 13N99 |
| spellingShingle |
polynomial ring irreducible polynomial Jacobian derivation 13N15; 13N99 Gavran, Volodimir Stepukh, Vitaliy On weakly semisimple derivations of the polynomial ring in two variables |
| topic_facet |
polynomial ring irreducible polynomial Jacobian derivation 13N15; 13N99 |
| format |
Article |
| author |
Gavran, Volodimir Stepukh, Vitaliy |
| author_facet |
Gavran, Volodimir Stepukh, Vitaliy |
| author_sort |
Gavran, Volodimir |
| title |
On weakly semisimple derivations of the polynomial ring in two variables |
| title_short |
On weakly semisimple derivations of the polynomial ring in two variables |
| title_full |
On weakly semisimple derivations of the polynomial ring in two variables |
| title_fullStr |
On weakly semisimple derivations of the polynomial ring in two variables |
| title_full_unstemmed |
On weakly semisimple derivations of the polynomial ring in two variables |
| title_sort |
on weakly semisimple derivations of the polynomial ring in two variables |
| description |
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 \(f_{1}(t), f_{2}(t)\in \mathbb K[t]\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1046 |
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AT gavranvolodimir onweaklysemisimplederivationsofthepolynomialringintwovariables AT stepukhvitaliy onweaklysemisimplederivationsofthepolynomialringintwovariables |
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2024-04-12T06:27:18Z |
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2024-04-12T06:27:18Z |
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