On divergence and sums of derivations
Let \(K\) be an algebraically closed field of characteristic zero and \(A\) a field of algebraic functions in \(n\) variables over \(\mathbb K\). (i.e. \(A\) is a finite dimensional algebraic extension of the field \(\mathbb K(x_1, \ldots, x_n)\) ). If \(\textit{D}\) is a \(\mathbb K\)-derivation...
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| Дата: | 2017 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2017
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/354 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-3542017-10-11T02:06:01Z On divergence and sums of derivations Chapovsky, E. Shevchyk, O. polynomial ring, derivation, divergence, jacobian derivation, transcendence basis Primary 13N15; Secondary 13A99, 17B66 Let \(K\) be an algebraically closed field of characteristic zero and \(A\) a field of algebraic functions in \(n\) variables over \(\mathbb K\). (i.e. \(A\) is a finite dimensional algebraic extension of the field \(\mathbb K(x_1, \ldots, x_n)\) ). If \(\textit{D}\) is a \(\mathbb K\)-derivation of \(A\), then its divergence \(div \textit{D}\) is an important geometric characteristic of \(\textit{D}\) (\(\textit{D}\) can be considered as a vector field with coefficients in \(A\)). A relation between expressions of \(div \textit{D}\) in different transcendence bases of \(A\) is pointed out. It is also proved that every divergence-free derivation \(\textit{D}\) on the polynomial ring \(\mathbb K[x, y, z]\) is a sum of at most two jacobian derivation. Lugansk National Taras Shevchenko University 2017-10-07 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/354 Algebra and Discrete Mathematics; Vol 24, No 1 (2017) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/354/pdf Copyright (c) 2017 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
polynomial ring derivation divergence jacobian derivation transcendence basis Primary 13N15; Secondary 13A99 17B66 |
| spellingShingle |
polynomial ring derivation divergence jacobian derivation transcendence basis Primary 13N15; Secondary 13A99 17B66 Chapovsky, E. Shevchyk, O. On divergence and sums of derivations |
| topic_facet |
polynomial ring derivation divergence jacobian derivation transcendence basis Primary 13N15; Secondary 13A99 17B66 |
| format |
Article |
| author |
Chapovsky, E. Shevchyk, O. |
| author_facet |
Chapovsky, E. Shevchyk, O. |
| author_sort |
Chapovsky, E. |
| title |
On divergence and sums of derivations |
| title_short |
On divergence and sums of derivations |
| title_full |
On divergence and sums of derivations |
| title_fullStr |
On divergence and sums of derivations |
| title_full_unstemmed |
On divergence and sums of derivations |
| title_sort |
on divergence and sums of derivations |
| description |
Let \(K\) be an algebraically closed field of characteristic zero and \(A\) a field of algebraic functions in \(n\) variables over \(\mathbb K\). (i.e. \(A\) is a finite dimensional algebraic extension of the field \(\mathbb K(x_1, \ldots, x_n)\) ). If \(\textit{D}\) is a \(\mathbb K\)-derivation of \(A\), then its divergence \(div \textit{D}\) is an important geometric characteristic of \(\textit{D}\) (\(\textit{D}\) can be considered as a vector field with coefficients in \(A\)). A relation between expressions of \(div \textit{D}\) in different transcendence bases of \(A\) is pointed out. It is also proved that every divergence-free derivation \(\textit{D}\) on the polynomial ring \(\mathbb K[x, y, z]\) is a sum of at most two jacobian derivation. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2017 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/354 |
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AT chapovskye ondivergenceandsumsofderivations AT shevchyko ondivergenceandsumsofderivations |
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2024-04-12T06:25:43Z |
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2024-04-12T06:25:43Z |
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1796109208854200320 |