Reducible fibers and monodromy of polynomial maps

Danny Neftin

For a polynomial f∈ℚ[x], Hilbert's irreducibility theorem asserts that the fiber f−1(a) is irreducible over ℚ for all values a∈ℚ outside a "thin" set of exceptions Rf. The problem of describing Rf is closely related to determining the monodromy group of f, and has consequences to arithmetic dynamics, the Davenport-Lewis-Schinzel problem, and to the polynomial version of the question: "can you hear the shape of the drum?". We shall discuss recent progress on describing Rf and its consequences to the above topics.   Based on joint work with Joachim König

https://www.ias.edu/video/reducible-fibers-and-monodromy-polynomial-maps