On closed rational functions in several variables

Details On closed rational functions in several variables On closed rational functions in several variables

2007-01-21

arxiv.org/abs/math/0701588v2

Mathematics

Rings and Algebras

Added references, corrected some typos

Scientific paper

Let k be an algebraically closed field of characteristic zero. An element F from k(x_1,...,x_n) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x_1,...,x_n). We prove that a rational function F=f/g is closed if f and g are algebraically independent and at least one of them is irreducible. We also show that the rational function F=f/g is closed if and only if the pencil af+bg contains only finitely many reducible hypersurfaces. Some sufficient conditions for a polynomial to be irreducible are given.

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