\input zb-basic
\input zb-ioport
\iteman{io-port 05320445}
\itemau{Bodin, Arnaud}
\itemti{Reducibility of rational functions in several variables.}
\itemso{Isr. J. Math. 164, 333-347 (2008).}
\itemab
Let $K$ be an algebraically closed field. Let $n\geq 2$ be an integer and $\underline{x}=(x_1, \ldots, x_n)$ be an $n-$tuple of algebraically independent variables (over $K$). Let $f=p/q \in K(\underline{x})$, with $p, q \in K[\underline{x}]$ such that $\text{gcd}(p,q)=1$; the degree of $f$ is $\deg f= \max(\deg p, \deg q)$. The rational function $f$ is composite if there exist $g \in K(\underline{x})$ and $r \in K(t)$ with $\deg r \geq 2$ such that $f(\underline{x})= r(g(\underline{x}))$. A consequence of a theorem of Bertini-Krull, is that: $f$ is non composite if and only if the number of $\lambda \in K \cup \{\infty\}$ such that $p- \lambda q$ is reducible in $K[\underline{x}]$, is finite. (By convention if $\lambda= \infty$ then $p- \lambda q= q$.) The main goal of the paper under review is to give a bound for this finite number. This number is $< (\deg f)^2 + \deg f$ for $n=2$ and characteristic zero (Theorem 4.1); and is $< (\deg f)^2 $ for $n\geq 2$ and any characteristic (Theorem 5.3). In fact this work is an extension of the polynomial case (i.e., $ f \in K[\underline{x}]$), but in this particular case such a bound is better, more precisely: if $ f \in K[\underline{x}]$ is non composite then the number of $\lambda \in K$ such that $f- \lambda$ is reducible in $K[\underline{x}]$ is $< \deg f.$ For a recent study of the polynomial case and for more references, we refer to [{\it S. Najib}, J. Algebra 292, No. 2, 566--573 (2005; Zbl 1119.13022)]. Reviewer's remarks: Recently some results are obtained in these directions by {\it A. Bodin, P. D\`ebes} and {\it S. Najib} in: [``Irreducibility of hypersurfaces", Commun. Algebra (2009)], and in: [``Indecomposable polynomials and their spectrum", Acta Arith. (2009)].
\itemrv{Salah Najib (Lille)}
\itemcc{}
\itemut{irreducible polynomial; composite polynomial; Bertini-Krull theorem; rational function}
\itemli{doi:10.1007/s11856-008-0033-2}
\end