Reducible Fibers of Polynomial Maps

2020/01/10 by Joachim König, König, Joachim, Danny Neftin +1 · 1 citation
Mathematics · #12E25 #20B15 #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.03630

openalex publication_date 2020/01/10 · openalex created_date 2022/09/11 · openalex updated_date 2026/07/28

Abstract

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=f₁∘⋯∘ f_r, as long as the indecomposable factors f_i∈\mathbb Q[x] are of degree at least 5 and are not xⁿ or a Chebyshev polynomial composed with linear polynomials.

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• Monodromy groups of polynomials of composition length 2

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