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Algebra and Discrete Mathematics, 2007, Issue 2, Pages 115–124
(Mi adm211)
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This article is cited in 2 scientific papers (total in 2 papers)
RESEARCH ARTICLE
On closed rational functions in several variables
Anatoliy P. Petravchuka, Oleksandr G. Ienaba a Kiev Taras Shevchenko University, Faculty of Mechanics and Mathematics, 64, Volodymyrska street, 01033 Kyiv,
Ukraine
b Kiev Taras Shevchenko University and
Technische Universität Kaiserslautern,
Fachbereich Mathematik, Postfach 3049,
67653 Kaiserslautern, Germany
Abstract:
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.
Keywords:
closed rational functions, irreducible polynomials.
Citation:
Anatoliy P. Petravchuk, Oleksandr G. Iena, “On closed rational functions in several variables”, Algebra Discrete Math., 2007, no. 2, 115–124
Linking options:
https://www.mathnet.ru/eng/adm211 https://www.mathnet.ru/eng/adm/y2007/i2/p115
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Abstract page: | 156 | Full-text PDF : | 54 | First page: | 1 |
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