On weakly semisimple derivations of the polynomial ring in two variables
Let K be an algebraically closed field of characteristic zero and K[x, y] the polynomial ring. Every element f ∈ K[x, y] determines the Jacobian derivation Df of K[x, y] by the rule Df(h) = detJ(f, h), where J(f, h) is the Jacobian matrix of the polynomials f and h. A polynomial f is called weakly s...
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| Опубліковано в: : | Algebra and Discrete Mathematics |
|---|---|
| Дата: | 2014 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/153346 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On weakly semisimple derivations of the polynomial ring in two variables / V.S. Gavran, V.V. Stepukh // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 50–58. — Бібліогр.: 7 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862713549552353280 |
|---|---|
| author | Gavran, V.S. Stepukh, V.V. |
| author_facet | Gavran, V.S. Stepukh, V.V. |
| citation_txt | On weakly semisimple derivations of the polynomial ring in two variables / V.S. Gavran, V.V. Stepukh // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 50–58. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Let K be an algebraically closed field of characteristic zero and K[x, y] the polynomial ring. Every element f ∈ K[x, y] determines the Jacobian derivation Df of K[x, y] by the rule Df(h) = detJ(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 Df(g) = λg for some nonzero λ ∈ 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 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) = f₁(x)f₂(y) for some f₁(t), f₂(t) ∈ K[t].
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| first_indexed | 2025-12-07T17:44:34Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-153346 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T17:44:34Z |
| publishDate | 2014 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Gavran, V.S. Stepukh, V.V. 2019-06-14T03:25:55Z 2019-06-14T03:25:55Z 2014 On weakly semisimple derivations of the polynomial ring in two variables / V.S. Gavran, V.V. Stepukh // Algebra and Discrete Mathematics. — 2014. — Vol. 18, № 1. — С. 50–58. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC:13N15; 13N99. https://nasplib.isofts.kiev.ua/handle/123456789/153346 Let K be an algebraically closed field of characteristic zero and K[x, y] the polynomial ring. Every element f ∈ K[x, y] determines the Jacobian derivation Df of K[x, y] by the rule Df(h) = detJ(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 Df(g) = λg for some nonzero λ ∈ 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 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) = f₁(x)f₂(y) for some f₁(t), f₂(t) ∈ K[t]. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics On weakly semisimple derivations of the polynomial ring in two variables Article published earlier |
| spellingShingle | On weakly semisimple derivations of the polynomial ring in two variables Gavran, V.S. Stepukh, V.V. |
| title | 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_short | On weakly semisimple derivations of the polynomial ring in two variables |
| title_sort | on weakly semisimple derivations of the polynomial ring in two variables |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/153346 |
| work_keys_str_mv | AT gavranvs onweaklysemisimplederivationsofthepolynomialringintwovariables AT stepukhvv onweaklysemisimplederivationsofthepolynomialringintwovariables |