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Reducible Fibers of Polynomial Maps

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.