This article is cited in 2 papers

RESEARCH ARTICLE

On closed rational functions in several variables

Anatoliy P. Petravchuk^a, Oleksandr G. Iena<sup>ba

a</sup> 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.

MSC: 26C15

Language: English

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