On closed rational functions in several variables

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.