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On closed rational functions in several variables. (English) Zbl 1164.26329
Summary: Let $${\mathbb K} = \overline{\mathbb K}$$ 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.

##### MSC:
 26C15 Real rational functions
##### Keywords:
closed rational function; irreducible polynomials