Reducible Fibers of Polynomial Maps

Abstract

Abstract For a degree $n$ polynomial $f\in {\mathbb {Q}}[x]$, the elements in the fiber $f^{-1}(a)\subseteq {\mathbb {C}}$ are of degree $n$ over ${\mathbb {Q}}$ for most values $a\in {\mathbb {Q}}$ by Hilbert’s irreducibility theorem. Determining the set of exceptional $a$’s without this property is a long standing open problem that is closely related to the Davenport–Lewis–Schinzel problem (1959) on reducibility of variable separated polynomials. As opposed to a previous work that mostly concerns indecomposable $f$, we answer both problems for decomposable $f=f_{1}\circ \cdots \circ f_{r}$, as long as the indecomposable factors $f_{i}\in {\mathbb {Q}}[x]$ are of degree $\geq 5$ and are not $x^{n}$ or a Chebyshev polynomial composed with linear polynomials.