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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.

26C15 Real rational functions