On closed rational functions in several variables
Let \(\mathbb K= \bar{\mathbb K}\) be a field of characteristic zero. An element \(\varphi\in \mathbb K(x_1,\dots, x_{n})\) is called a closed rational function if the subfield \(\mathbb K(\varphi)\) is algebraically closed in the field \(\mathbb K(x_1,\dots, x_{n})\). We prove that a rational funct...
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| Date: | 2018 |
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| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/848 |
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oai:ojs.admjournal.luguniv.edu.ua:article-8482018-03-21T11:59:09Z On closed rational functions in several variables Petravchuk, Anatoliy P. Iena, Oleksandr G. closed rational functions, irreducible polynomials 26C15 Let \(\mathbb K= \bar{\mathbb K}\) be a field of characteristic zero. An element \(\varphi\in \mathbb K(x_1,\dots, x_{n})\) is called a closed rational function if the subfield \(\mathbb K(\varphi)\) is algebraically closed in the field \(\mathbb K(x_1,\dots, x_{n})\). We prove that a rational function \(\varphi=f/g\) is closed if \(f\) and \(g\) are algebraically independent and at least one of them is irreducible. We also show that a rational function \(\varphi=f/g\) is closed if and only if the pencil \(\alpha f+\beta g\) contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are given. Lugansk National Taras Shevchenko University 2018-03-21 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/848 Algebra and Discrete Mathematics; Vol 6, No 2 (2007) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/848/378 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-03-21T11:59:09Z |
| collection |
OJS |
| language |
English |
| topic |
closed rational functions irreducible polynomials 26C15 |
| spellingShingle |
closed rational functions irreducible polynomials 26C15 Petravchuk, Anatoliy P. Iena, Oleksandr G. On closed rational functions in several variables |
| topic_facet |
closed rational functions irreducible polynomials 26C15 |
| format |
Article |
| author |
Petravchuk, Anatoliy P. Iena, Oleksandr G. |
| author_facet |
Petravchuk, Anatoliy P. Iena, Oleksandr G. |
| author_sort |
Petravchuk, Anatoliy P. |
| title |
On closed rational functions in several variables |
| title_short |
On closed rational functions in several variables |
| title_full |
On closed rational functions in several variables |
| title_fullStr |
On closed rational functions in several variables |
| title_full_unstemmed |
On closed rational functions in several variables |
| title_sort |
on closed rational functions in several variables |
| description |
Let \(\mathbb K= \bar{\mathbb K}\) be a field of characteristic zero. An element \(\varphi\in \mathbb K(x_1,\dots, x_{n})\) is called a closed rational function if the subfield \(\mathbb K(\varphi)\) is algebraically closed in the field \(\mathbb K(x_1,\dots, x_{n})\). We prove that a rational function \(\varphi=f/g\) is closed if \(f\) and \(g\) are algebraically independent and at least one of them is irreducible. We also show that a rational function \(\varphi=f/g\) is closed if and only if the pencil \(\alpha f+\beta g\) contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are given. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/848 |
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AT petravchukanatoliyp onclosedrationalfunctionsinseveralvariables AT ienaoleksandrg onclosedrationalfunctionsinseveralvariables |
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2025-07-17T10:35:49Z |
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2025-07-17T10:35:49Z |
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1837890063986327553 |