Reducible Fibers of Polynomial Maps

For a degree n polynomial f ∈ Q[x], the elements in the fiber f⁻¹(a) ⊆ C are of degree n over Q for most values a ∈ 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 = f₁ ◦ · · · ◦ f_r, as long as the indecomposable factors f_i ∈ Q[x] are of degree ≥ 5 and are not xⁿ or a Chebyshev polynomial composed with linear polynomials.