Reducible Fibers of Polynomial Maps

Abstract

For a degree $n$ polynomial $f$ over the rationals, the elements in the fiber $f^{-1}(a)$ are of degree $n$ over $\mathbb{Q}$ for most rational values $a$ 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 separated polynomials. As opposed to previous work which 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 at least $5$ and are not $x^n$ or a Chebyshev polynomial composed with linear polynomials.

26 pgs. The manuscript was shortened to focus on polynomials and makes clearer the use of the classification of finite simple groups