2007/01/21 by А. П. Петравчук, A. P. Petravchuk, Petravchuk, A. P. +3 · 1 citation
Computer Science · Mathematics · #26C15 #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #FOS: Mathematics #Functional Equations Stability Results #Polynomial and algebraic computation #Rings and Algebras (math.RA) #math.AC #math.RA #msc:26C15
paper · pdf · doi:10.48550/arxiv.math/0701588
Added references, corrected some typos
openalex publication_date 2007/01/21 · arxiv created 2007/02/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let k be an algebraically closed field of characteristic zero. An element F from k(x1,...,xn) is called a closed rational function if the subfield k(F) is algebraically closed in the field k(x1,...,xn). 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.