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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 ) 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 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 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...

Stable reduction of three point covers

Stefan Wewers (2005)

Journal de Théorie des Nombres de Bordeaux

This note gives a survey of some recent results on the stable reduction of covers of the projective line branched at three points.

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