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Article
Specializations of one-parameter families of polynomials
ANNALES DE L INSTITUT FOURIER
  • F Hajir, University of Massachusetts - Amherst
  • S Wong
Publication Date
2006
Abstract

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 of the modular polynomial Φ n (x,t) (which vanishes on the j-invariants of pairs of elliptic curves related by a cyclic n-isogeny), and show that for any n≥53, all but finitely many of the K-specializations of Φ n (x,t) are K-irreducible and have Galois group containing SL 2 (ℤ/n)/{±I}. Third, for a simple branched cover π:Y→ℙ K 1 of degree n≥7 and of genus at least 2, all but finitely many K-specializations are K-irreducible and have Galois group S n .

Comments

This is the pre-published version harvested from ArXiv. The published version is located at http://aif.cedram.org/item?id=AIF_2006__56_4_1127_0

Pages
1127-1163
Citation Information
F Hajir and S Wong. "Specializations of one-parameter families of polynomials" ANNALES DE L INSTITUT FOURIER Vol. 56 Iss. 4 (2006)
Available at: http://works.bepress.com/farshid_hajir/4/