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 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=f1·s fr, as long as the indecomposable factors fi∈ Q[x] are of degree at least 5 and are not xn or a Chebyshev polynomial composed with linear polynomials.
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