On the reducibility of fibers of polynomials

Given a polynomial f in Q[x], by Hilbert's irreducibility theorem, the set of a in Q such that the fiber \(f^{-1}(a)\) is reducible over Q is thin. In this talk, we discuss recent progress regarding the following two closely related problems that go back at least to the 50's.

Problem 1. Given a polynomial f in Q[x], detemine the set of all integers a in Z such that \(f^{-1}(a)\) is reducible over Q.
Problem 2. (Davenport-Lewis-Schinzel problem) For which polynomials f and g in C[x]\C, is the polynomial f(x)-g(y) in C[x,y] reducible?

In fact, these two problems are solved unless f factors through an indecomposable polynomial of small degree. This is based on a joint work with Danny Neftin.