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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| Видавець: | Інститут прикладної математики і механіки НАН України |
|---|---|
| Дата: | 2014 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2014
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/153346 |
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| 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. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1533462019-06-15T01:26:25Z On weakly semisimple derivations of the polynomial ring in two variables Gavran, V.S. Stepukh, V.V. 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]. 2014 Article 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. http://dspace.nbuv.gov.ua/handle/123456789/153346 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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]. |
| format |
Article |
| author |
Gavran, V.S. Stepukh, V.V. |
| spellingShingle |
Gavran, V.S. Stepukh, V.V. On weakly semisimple derivations of the polynomial ring in two variables Algebra and Discrete Mathematics |
| author_facet |
Gavran, V.S. Stepukh, V.V. |
| author_sort |
Gavran, V.S. |
| 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 |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/153346 |
| 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. |
| series |
Algebra and Discrete Mathematics |
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2023-05-20T17:38:31Z |
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2023-05-20T17:38:31Z |
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1796153750119776256 |