DATE:      3/31/05
TIME:      12:10 pm in Weber 117
SPEAKER:   Siman Wong, (UMass Amherst)
TITLE:     Specializations of one-parameter families of polynomials

ABSSTRACT: Let K be a number field, and let \lambda(x, t) in K[x, t]
be irreducible over K(t). Using algebraic geometry and group theory,
we study the set of \alpha in K for which the specialized polynomial
\lambda(x, \alpha) is K-reducible. We apply this to show that for any
fixed n >= 10 and for any number field K, all but finitely many
K-specializations of the degree n generalized Laguerre polynomial are
K-irreducible and have Galois group Sn. In conjunction with the theory
of complex multiplication, we also show that for any K and for any 
n >= 53, all but finitely many of the K-specializations of the modular
equation \Phin(x, t) are K-irreducible and have Galois group
containing PSL2(\Z/n).