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Specializations of one-parameter families of polynomials

Farshid Hajir; Siman Wong — 2006

Annales de l’institut Fourier

Let $K$ be a number field, and suppose $λ (x, t) \in K [x, t]$ is irreducible over $K (t)$ . Using algebraic geometry and group theory, we describe conditions under which the $K$ -exceptional set of $λ$ , i.e. the set of $α \in K$ for which the specialized polynomial $λ (x, α)$ is $K$ -reducible, is finite. We give three applications of the methods we develop. First, we show that for any fixed $n \geq 10$ , all but finitely many $K$ -specializations of the degree $n$ generalized Laguerre polynomial $L_{n}^{(t)} (x)$ are $K$ -irreducible and have Galois group $S_{n}$ . Second, we study specializations...

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