Diversity in parametric families of number fields

Abstract

Let X be a projective curve defined over Q and t ε Q(X) a non-constant rational function of degree v > 2. For every n ε Z pick such that t(Pn) = n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P1),...,Q(PN) there are at least cN/log N distinct; here, c > 0 depends only on the degree v and the genus g = g(X). We prove that there are at least N/(log N)1-η distinct fields, where η > 0 depends only on v and g.